^ A TREATISE ON ^ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS BY WILLIAM WOOLSEY JOHNSON Professor of Mathematics at the United States Naval Academy Annapolis Maryland THIRD E^PITION SECOND THOUSAND. NEW YORK: JOHN WILEY & SONS, 53 East Tenth Street. 1893. Entered according to Act of Congress, in the year 1889, by WILLIAM WOOLSEY JOHNSON, in the Office of the Librarian of Congress, at Washington. PREFACE. The treatment of the subject of Differential Equations here presented will, it is hoped, be found complete in all those portions which bear upon their practical applications, and in the discussion of their theory so far as it can be adequately treated without the use of the complex variable. The topics included and the order pursued are sufficiently indicated by the table of contents. An amount of space somewhat greater than usual has been devoted to the geometrical illustrations which arise when the variables are regarded as the rectangular coordinates of a point. This has been done in the belief that the conceptions pecuHar to the subject are more readily grasped when embodied in their geometric representations. In this connection the sub- ject of singular solutions of ordinary differential equations and the conception of the characteristic in partial differential equa- tions may be particularly mentioned. Particular attention has been paid to the development of symbolic methods, especially in connection with the operator X — , for which, in accordance with recent usage, the symbol i9 ax has been adopted. Some new applications of this symbol have been made. 242511 IV PREFACE. The expression "binomial equations" is applied in this work (in a sense introduced by Boole) to those linear equations which are included in the general form f^{ff)y j^ x%{f^)y =.0^ and which constitute the class of equations best adapted to solution by development in series. In the sections treating of this method a uniform process has been adopted for the secondary or logarithmic solutions which occur in certain cases. The development of the particular integral when the second member is a power of x is also considered. Chapter VIII is devoted to the general solution of the binomial equation in the notation of the hypergeometric series, and Chapter IX to Riccati's, Bessel's and Legendre's equations. The examples at the ends of the sections have been derived from various sources, and not a few prepared expressly for this work. They are arranged in order of difficulty, and the solu- tions are given. These have been verified in the proof-sheets, so that it is believed that they will be found free from errors. The ordinary references in the text are to Rice and John- son's Diff. Calc. and Johnson's Int. Calc, published by John Wiley and Sons uniformly with the present volume. W. W.J. U. S. Naval Academy, May, 1889. CONTENTS, CHAPTER I. NATURE AND MEANING OF A DIFFERENTIAL EQUATION BETWEEN TWO VARIABLES. I. PAGE Solutions in the Form y = Fix) i Solutions not in the Form y — Fix) , 3 Particular and Complete Integrals 3 Primitive of a Differential Equation 5 Number of Arbitrary Constants 6 Geometrical Illustration of the Meaning of a Differential Equation 8 Systems of Curves containing an Arbitrary Parameter 9 Doubly Infinite Systems of Curves 10 Examples 1 12 CHAPTER 11. EQUATIONS OF THE FIRST ORDER AND DEGREE. II. Separation of the Variables 14 Reduction of the Integral to Algebraic Form 15 Homogeneous Equations 16 Similar and Similarly situated Systems of Curves 18 Case in which the Coefficients of dx and dy are of the First Degree 19 Examples II 20 III. Exact Differential Equations 22 Integrating Factors 24 Vi CONTENTS. PAGB Expressions of the Form x<^y^(jnydx + nxdy) 27 Examples III 29 IV. The Linear Equation of the First Order 30 Transformation of a Differential Equation 32 Extension of the Linear Equation 33 Examples IV 34 CHAPTER III. EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE. V. Decomposable Equations 37 Equations Properly of the Second Degree 38 Systems of Curves corresponding to Equations of Different Degrees 39 Standard Form of the Integral of an Equation of the Second Degree 42 Singular Solutions 43 The Discriminant 45 Cusp-Loci 46 Tac-Loci and Node-Loci 47 Examples V 50 VI. Solution by Differentiation 52 Equations from which One of the Variables is Absent 54 Homogeneous Equations not of the First Degree in / 57 The Equation of the First Degree in x and ^ 58 Clairaut's Equation 59 Examples VI 61 VII. Geometrical Applications 63 Polar Coordinates 64 The required Curve a Singular Solution 66 Orthogonal Trajectories 67 Examples VII 69 CONTENTS. vii CHAPTER IV. EQUATIONS OF THE SECOND ORDER. VIII. Successive Integration 72 The First Integrals 74 Integrating Factors .y. 76 Singular Solutions of Equations of the Second Order (foot-note) 77 Derivation of the Complete Integral from Two First Integrals 78 Exact Equations of the Second Order 80 Equations in which y does not occur 82 Equations in which x does not occur yj, 83 The Method of Variation of Parameters 84 Examples VIII 87 CHAPTER V. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS. IX. Properties of the Linear Equation 91 The Linear Equation with Constant Coefficients and Second Member Zero. ... 93 Case of Equal Roots 95 Case of Imaginary Roots 97 The Linear Equation with Constant Coefficients and Second Member a Func- tion oi X 98 The Inverse Operative Symbol 99 General Expression for the Integral loi Examples IX 103 X. Symbolic Methods of Integration 106 The Second Member X of the Form e^x 106 Case in which X contains a Term of the Form %\w ax ox co^ ax 109 Case in which X contains Terms of the Form x*n 112 Symbolic Formula of Reduction for the Form e^x V. 114 Application to the Evaluation of an Ordinary Integral (see also p. 1 18) 116 Symbolic Formula of Reduction for the Form xV. 116 Symbolic Formula of Reduction for the Form xrV 118 Employment of the Exponential Forms of sin ax and cos ajr 119 Examples X 120 Vlll CONTENTS. CHAPTER VI LINEAR EQUATIONS WITH VARUBLE COEFnCIENTS. XI. PAGE The Homogeneous Linear Equation 12^ The Operative Symbol tV Complete Integral of the Equation f(^H)y = o 127 Cases of Equal and Imaginary Roots 127 The Particular Integral oif{fi)y -X .!..!!! 128 Case in which X is of the Form x<* ,2g Symbol Solutions of Linear Equations with Variable Coefficients 130 Non-Commutative Symbohc Factors j ^2 Examples XI . _^ ' ■ " *34 XII. Exact Linear Equations j ^c The Condition of Direct Integrability i ^y ^ Integrating Factors of the Form x^ , ^o Symbolic Treatment of Exact Linear Equation 140 Symbolic Formulae involving D and »? ,41 Examples XII j XIII. The Linear Equation of the Second Order 14- Case in which an Integral ji, when the Second Member is Zero, is known 147 Expression for the Complete Integral in Terms of jf/j 149 Relation between Two Independent Integrals ji and^/a 151 Symmetrical Expression for the Particular Integral i r2 Resolution of the Operator into Factors ic^ The Related Equation of the First Order i r^ The Transformation y = v/(x) jc- The Transformation y = eo^"*v jcy Removal of the Term containing the First Derivative — the Normal Form .... 158 The Invariant for the Transformation y = v/(x) (see also Ex. 26, p. 165) 159 Change of the Independent Variable i5o Examples XIII j52 CONTENTS. ix CHAPTER VII. SOLUTIONS IN SERIES. ^ XIV. PAGE Development of the Integral of a Differential Equation in Series i66 Development of the Independent Integrals of a Linear Equation whose Second Member is Zero 167 Convergency of the Series 171 Development of the Particular Integral 172 Binomial and Polynomial Equations 1 73 Finite Solutions 1 74 Examples XIV 177 XV. Development of the Logarithmic Form of the Second Integral — Case of Equal Values of w 181 Case in which the Values of m differ by a Multiple of j 185 Special Forms of the Particular Integral 191 Examples XV 194 CHAPTER VIIL THE HYPERGEOMETRIC SERIES. XVL General Solution of the Binomial Equation of the Second Order 198 Differential Equation of the Hypergeometric Series 201 Integral Values of 7 and 7' 202 The Supplementary Series when y^, is a Finite Series 204 Imaginary Values of o and fi 205 Infinite Values of a and ^ 206 Case in which a or j8 equals 7 or Unity 208 The Binomial Equation of the Third Order 209 Development of the Solution in Descending Series 210 Transformation of the Equation of the Hypergeometric Series 211 Change of the Independent Variable 214 The Twenty-Four Integrals 216 Solutions in Finite Form 218 Examples XVI 220 CONTENTS. >/. CHAPTER IX. SPECIAL FORMS OF DIFFERENTIAL EQUATIONS. XVII. PAGE Riccati's Equation 224 Standard I^inear Form of the Equation 225 Finite Solutions 228 Relations between the Six Integrals 230 Transformations of Riccati's Equation 232 Bessel's Equation 234 Finite Solutions 235 The Besselian Functions • • . . 237 The Besselian Functions of the Second Kind 239 Legendie's Equation 241 The Legendrean Coefficients 243 The Second Integral Qn when n is an Integer 244 Examples XVII 247 CHAPTER X. EQUATIONS INVOLVING MORE THAN TWO VARIABLES. XVIII. Determinate Systems of the First Order 25 1 Transformation of Variables 252 Exact Equations 253 The Integrals of a System 254 Equations of Higher Orders equivalent to Determinate Systems of the First Order 256 Geometrical Meaning of a System involving Three Variables 257 Examples XVIII 258 XIX. Simultaneous Linear Equations 260 Number of Arbitrary Constants 263 Introduction of a New Variable 264 Examples XIX 266 CONTENTS. xi XX. PAGE Single Differential Equations involving more than Two Variables 270 The Condition of Integrability 270 Solution of the Integrable Equation 272 Separation of the Variables 274 Homogeneous Equations 275 Equations containing more than Three Variables 276 The Non-Integrable Equation 278 Monge's Solution - 280 Geometrical Meaning of a Single Differential Equation between Three Variables 281 The Auxiliary System of Lines 282 Distinction between the Two Cases 283 Examples XX 284 CHAPTER XI. PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. XXI. Equations involving a Single Partial Derivative 287 Equations of the First Order and Degree 288 Lagrange's Solution 289 Geometrical Illustration of Lagrange's Solution 291 Orthogonal Surfaces 292 The Complete and General Primitives 293 Derivation of the Differential Equation from the General Primitive 294' Examples XXI 297 XXII. The Non-Linear Equation of the First Order 299 The System of Characteristics 3°° The General Integral 3^4 Derivation of a Complete Integral from the Equations of the Characteristic 306 Relation of the General to the Complete Integral 309 Singular Solutions 3 ^ ^ Integrals having Single Points of Contact with the Singular Solution 312 Derivation of the Singular Solution from the Differential Equation 313 Equations involving p and q only 3^4 Integrals formed by Characteristics passing through a Common Point 314 Equation Analogous to Clairaut's 3^5 Equations not containing x ox y ". 3^^ XU CONTENTS. PAGB Equations of the Form fi{x,p) =f-i{y, q) 317 Change of Form in the Equations of the Characteristic 319 Transformation of the Variables 321 Examples XXII 323 CHAPTER XXL PARTIAL DIFFERENTIAL EQUATIONS OF HIGHER ORDER. xxiir. Equations of the Second Order 326 The Primitive containing Two Arbitrary Functions 326 Forms which give Rise to Equations of the Second Order 327 The Intermediate Equation of the First Order 329 Successive Integration 330 Monge's Method 332 Integrability of Monge's Equations ■^■^z^ Illustrative Examples 33 e Examples XXIII 340 XXIV. Linear Equations 341 >mogeneous Equations with Constant Coefficients 342 ^ .nbolic Solution of the Component Equations of the Form (/) — mD')z = o. . 344 Case of Equal Roots 34r Case of Imaginary Roots 347 The Particular Integral 348 The Second Member of the Form 4»(a;r + by) 350 The Non-Homogeneous Equation 3C2 Special Forms of the Integral 354 Special Methods for the Particular Integral 355 The Second Member of the Form e"^ + h 3^6 The Second Member of the Form sm{ax + by) or cos{ax -{• by) 357 The Second Member of the Form x^y^ 358 The Second Member of the Form e^^ + hy 360 Linear Equations with Variable Coefficients 361 The Equation F{fy, fy')z = , 363 The Equation eIj% //'> =V 364 The Symbol »9 + ^9' in Relation to the Homogeneous Function of x and y . . . . 365 Examples XXIV 366 DIFFERENTIAL EQUATIONS, CHAPTER I. NATURE AND MEANING OF A DIFFERENTIAL EQUATION BETWEEN TWO VARIABLES. I. Solutions in the Form y = F{x). I. In the Integral Calculus, we suppose the differential of a variable y to be given in terms of another variable x and ii ' differential dx, and we seek to express y as a function of x ; in other words, since we know that the form of the given equation must be dy = f{x)dx, ........ (i) which may be written l = /(^>' <^> the derivative of j/ is given in terms of x^ and an equation of the form J = ^W (3) is said to satisfy the given equation (i) or (2) when F{x) is a function whose derivative is the given function f{x). I 2 DIFFERENTIAL EQUATIONS. [Art. 2. 2. A differential equation between two variables x and y is an equation involving in any manner one or more of the derivatives of the unknown function y with respect to Xy together with one or both of the variables x and y. The order of the equation is that of the highest derivative contained in it, and its degree is that of the highest power of this deriva- tive which occurs. An equation of the form y = F{x) satisfies the differential equation if the substitution of F{x) and its derivatives for y and its derivatives reduces it to an identity. For example, the differential equation dx^ dx will be found on trial to be satisfied by j/ = ^-^ sin x ; for, if we substitute this value for j, and for its derivatives the resulting values -^ =■ ^(cos x 4- sin x) and — ^ = 2e^ cos x, the first dx dx^ member reduces to zero. 3. Equation (i) is, in fact, the simplest form of differential equation. Its general. solution is expressed by the formula = f/(^ )dx', (4> and it is the province of the Integral Calculus to reduce this expression, when possible, to a form free from the integral sign, and involving only known functional symbols. But, when this is not possible, the second member of equation (4) represents a new functional form, which, by definition, satisfies equation (i) ; so that the formula is still the solution of the differential equa- tion. In like manner, a differential equation of any other form is said to be solved when a proper expression is found, even though it involve integrals which we are unable to reduce. § I.] PARTICULAR AND COMPLETE INTEGRALS. Solutions not in the Form y = F{oc). 4. A relation between x and y not in the form y =. f{x) may satisfy a differential equation. When this is the case, the values of the derivatives employed in verifying will be expressed in terms of x and y ; and, when these are substituted in the differential equation, the result is a relation between x and y which should be true in virtue of the integral equation. For example, in order to show that the equation KIT-^l+^=° ('> is satisfied by f = 4^^, (2) we differentiate equation (2) ; thus, and, substituting the value of -^ from (3) in equation (i), we Q/X have f 4^^ X- 2^ + ^ = o* This equation is true by virtue of the integral relation (2) ; equation (2) is therefore a solution of the given differential equation (i). Particular and Complete Integrals. 5. If Fix) is a particular value of the integral in equation (4), Art. 3, then y = F{x) + C, 4 DIFFERENTIAL EQUATIONS. [Art. 5. where 6^ is a constant to which any value may be assigned, is the general or complete solution of equation (i). Thus, the general solution involves an arbitrary constant which is called the constant of integration. In like manner, a solution of any differential equation is called a particular integral of the equa- tion ; but the most general solution, which is called the complete integral, contains one or more arbitrary constants of integration, the manner in which these constants enter the equation depend- ing on the form of the differential equation. For example, it was noticed in Art. 2 that the differential equation jy 2-^ + 2y = o (i) ax^ ax is satisfied by y ■= e'^smx, (2) which is, therefore, a particular integral. It is not difficult, in this case, to infer from this solution the complete integral ; for, in the first place, it is evident that, if we multiply the value of y given in equation (2) by the constant C, the values of ^ and ^-^ will also be multiplied by C, so that the result of dx dx^ substitution in the first member of equation (i) will be C times the previous result, and therefore still equal to zero. Thus, y = C^^sin.^ (3) is a more general solution of the differential equation. Again, since x does not explicitly enter equation (i), and -r + a, where a is a constant, has the same differential as x, ^ = CV^ + « sin (.:*: + a) (4) satisfies the equation, and forms a still more general solution. § I.] PRIMITIVE OF A DIFFERENTIAL EQUATION. 5 Expanding sin i^x + a), and putting CV^cosa = A^ C^^sina = By we may write the solution in the form y = Ae^^YCix 4- Be^CQ^x, (5) in which A and B are two independent arbitrary constants, because C and a are independently arbitrary. We shall see presently that this equation containing two arbitrary constants is the complete integral of equation (i). The particular integral (2) is the result of putting A = i and ^ = o in the complete integral. Primitive of a Differential Equation. 6. The general solution found in the preceding article may, of course, be verified by the substitution of the values of j^, -^, and -^ in the differential equation. Thus, from dx dx^ ^ y = Ae^sinx 4- ^^^cos^, .......... (i) we get -i' = Ae^(smx + cos^) + Be^{cosx — sin;\?),. . (2) ax and -— ■=■ 2Ae^Q.o's>x — 2Be^smx; (3) dx^ andy if these values are substituted in the first member of ^ ,f+,y=0, ..... . (4) dx^ ax the coefficients of A and B separately reduce to zero, and the equation .is satisfied independently of the values of A and B. 6 DIFFERENTIAL EQUATIONS, [Alt. 6. It thus appears that the differential equation (4) is the same as the result of eliminating A and B from equations (i), (2), and (3). Equation (i) is, in this point of view, said to be the primi- tive of equation (4). 7. So also any equation containing arbitrary constants is the primitive of a certain differential equation free from those constants. For example, if, in the equation c^x — cy -\r a — o, (i) c is regarded as an arbitrary constant, we have, by differentia- tion, ^-^:7 = o, or / = <^; (2) ax dx whence, eliminating c, we obtain -©■- 't*'-" « as the equation of which equation (i) is the primitive. Again, equation (2), from which a has disappeared by differ- entiation, is itself the equation derived from equation (i) as a primitive, when a is regarded as an arbitrary, and ^ as a fixed, constant. But, if both a and c are arbitrary, differentiating again, we have dx' and, c having disappeared, this is the equation of the second order of which equation (i) is the primitive. . It is evident that, in every case, the number of differentia- tions necessary, and therefore the index of the order of the differential equation produced, will be the same as the number of constants to be eliminated. § L] NUMBER OF ARBITRARY CONSTANTS. 7 8. Considering now the differential equation as given, the primitive is an integral equation which satisfies it, the constants eliminated being, in the reverse process of finding the integral, the constants of integration ; and it is the most general solution, or complete integral, because no greater number of constants could be eliminated without introducing derivatives higher than the highest which occurs in the given equation. For example, the process given in the preceding article shows that c^x — cy -{- a = o is the complete integral of ^\dx) dy , y-f -^ a — o. ax It was shown in Art. 4 that this differential equation is satisfied by y'^ = 4ax, which, it will be noticed, is not a particu- lar case of the complete integral. Thus, while the complete integral is the most general solution, it. does not, in all cases, include all the solutions. 9. We thus see that the complete integral of a differential equation of the first order should contain one constant of integration, that of an equation of the second order should contain two constants, and so on. It is, of course, to be under- stood that no two of the constants admit of being replaced by a single one. For example, the constants C and a in the equa- tion y = C^^ + " are equivalent to a single arbitrary constant ; for, putting A = CV", the equation may be written y = Ae^, hence it is the complete integral of an equation of the first, not of the second, order. DIFFERENTIAL EQUATIONS. [Art. 10. Geometrical Illustration of the Meaning of a Differential Equation. 10. Let X and ^ in a differential equation be regarded as the rectangular coordinates of a point in a plane ; then the derivative -^ is the tangent of the inclination to the axis of x dx of the direction in which the point (jir, y) is moving. Putting / = ^dy dx' a differential equation of the first order is a relation between the variables Xy y^ and /, of which x and y determine the position of the point, and / the direction of its motion. We may assign to X and y any values we choose, and then determine from the equation one or more values of /. We cannot, therefore, regard the differential equation as satisfied by certain points (that is, by certain associated values of x and y) ; but it is satisfied by certain associated values of x, j/, and /, that is, by a point in any position, provided it is moving in the proper direction. II. Let us now suppose the point {x, y) to start from any assumed initial position, and to move in the proper direction. We have thus a moving point satisfying the differential equa- tion. As the point moves, the values of x and y vary, so that the value of / derived from the equation will likewise, in gen- eral, vary ; and we may suppose the direction of the point's motion to vary in such a way that the moving point continues to satisfy the differential equation. The line which the pomt now describes is, in general, a curve ; and the point may evi- dently move along this curve in either direction, and yet always satisfy the differential equation. The moving point may return to its initial position, thus describing a closed curve ; or it may pass to infinity in both directions, describing an infinite branch of a curve. § I.] GEOMETRICAL REPRESENTATION. 9 If, now, we can determine the equation of this curve in the form of an ordinary equation between x and j, the value of -^ dx found by differentiating the equation of the curve will, by hypothesis, be identical, for any values of x and y^ with the value of / corresponding to the same values of x and y in the differential equation. The equation of the curve will, there- fore, be a solution, or integral, of the differential equation. 12. But, since this integral equation restricts the point to certain positions, the assemblage of which constitutes the curve, it is not the complete solution of the differential equa- tion ; for the complete solution ought to represent the moving point satisfying the equation in all its possible positions. If, now, we take for initial point any point not on the curve already determined, and proceed in like manner, we shall determine another curve, whose equation will be another particular solu- tion, or integral, of the differential equation. We thus have an unlimited number of curves forming a system of curves, and the complete integral is the general equation of this system. This general equation must contain, besides x and y, a. quantity independent of x and y called the parameter^ by giving different values to which we obtain the equations of all the particular curves of the system. The arbitrary parameter of the system is, of course, the constant of integration. 13. We may illustrate this by a simple example. Let the differential equation be . 1"=-^ (■) ax y y , Since — is the tangent of the inclination to the axis of x of the line joining the point {x, y) to the origin, the equation expresses that the point (x, y) is always moving in a direction perpendicu- lar to the line joining it to the origin. Starting from any initial 10 DIFFERENTIAL EQUATIONS. [Art. 1 3. position, it is clear that the point describes a circle about the origin as centre. The system of curves in this case, therefore, consists of all circles whose centres are at the origin ; and the general equation of this system, x^ ^ f ^C, (2) where C is the parameter, is the complete integral. Now consider the moving point when in any special posi- tion, as, for instance, in the position (3, 2) ; we find, by substi- tuting these values for x and y in equation (2), C = 13. Hence ^^ + / = 13 is the equation of the particular curve in which the point is then moving. If we differentiate this equation, we find a value for -^ at the point (3, 2) identical with that given for the same ax point by equation (i). Doubly Infinite Systems of Curves. 14. In the case of a differential equation of the second order, let -^ — p and -^ — Q'i then the equation is a relation between x, y, p, and q. It is possible to assign any values we please to x, y, and /, and to determine from the equation a value of q, which, in connection with the assumed values of x, y, and /, will satisfy the equc This value of q, in connection with the assumed value determines the curvature of the path of the moving ection \ § I.] GEOMETRICAL representation: II (;r, y). Hence a differential equation of the second order may be regarded as satisfied by a moving point having any assumed position, and moving in any assumed direction, provided only that its path have the proper curvature. Starting from any assumed initial point, and in any assumed initial direction, the point {Xy y) may move in such a manner as to satisfy the equa- tion. As it moves the values of x and y will vary ; and, since the path has a definite curvature at this point, the value of / will likewise vary. Hence the value of q derived from the differential equation will, in general, also vary ; but we may suppose the curvature of the path to vary in such a manner that the moving point continues to satisfy the equation. A curve is thus described whose ordinary equation is a solution of the differential equation, since the simultaneous values of x, y, -^, and — =^, at every point of it, by hypothesis, satisfy that equation. 15. As before, the complete integral is the general equation of the system of curves which may be generated in the manner explained above ; but this system has a greater generality than that which represents a differential equation of the first order. For, in its general equation, it must be possible to assign any assumed simultaneous values to x^ y, and /. Substituting the assumed values in the general equation and in the result of its differentiation, we have two equations ; and, in order to satisfy them, we must have two arbitrary parameters at our disposal. The system of curves representing a differential equation of the second order is, therefore, a system containing two param- eters, to each of which independently an unlimited number of values may be assigned. Such a system is said to be a doubly ittfinite system of curves. In like manner, it may be shown that a differential equation of the third order is represented by a triply infinite system of curves, and so on. 12 DIFFERENTIAL EQUATIONS. [Art. 1 5. Examples I. 1. Form the differential equation of which >' = ^cos a: is the com- plete integral. -^ + y\2Xix = o. ax 2. Form the equation of which y = ax^ + bx is the complete inte- gral, a and b being arbitrary. , d^y dy , XT — =^ — 2X-^ -f- 2y = o. dx' dx 3. Form the equation of which y^ — 2cx — ^r^ = o is the complete integral. H- 2X-^ — y = o. dx it) 4. Form the equation of which ^^^ -f- 2cxey -f- ^^ _ q jg ^j^g primitive. 5. Form the equation of which y = {x + c)e^^ is the complete integral. -^ = (?«-^ + ay. dx > 6. Denoting by B the inclination to the axis of x of the line joining {x, y) to the origin, and by ^ the inclination of the point's motion, write the differential equation which expresses that ^ is the supplement of B, and show that it represents a system of hyperbolas. 7. With the same notation, write the differential equation which expresses that "<^ = 2B, and show that it represents a system of circles passing through the origin. § L] EXAMPLES. 13 8. Form the, differential equation of the system of straight lines which touch the circle ^r^ -j^ jv^ = i, and show that this circle also satisfies the equation. 9. Find the differential equatioh of all the circles having their radii equal to a. i'.(i)"-j=-(g)' 10. Find the differential equation of all the conies whose axes coincide with the coordinate axes. •^ dx \dx) ^ dx'' 14 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 1 6. THAPTER II. EQUATIONS OF THE FIRST ORDER AND DEGREE. 11. Separation of the Variables, i6. In an equation of the first order, it is immaterial whether X or j/hQ taken as the independent variable. If the equation is also of the first degree, it is frequently written in the form Mdx + Ndy — o, in which M and N denote functions of x and y. The simplest case is that in which the equation may be so writter^ that the coefficient of dx is a function of x only, and that oi j/ a, function of jj/ only ; in other words, the case in which the equation can be written in the form /(x)dx + {y)dy = o (i) The complete integral is then evidently ^/{x)dx+^cl>iy)dy = C. (2) § IL] SEPARATION OF THE VARIABLES. 1 5 17. The process of reducing an equation, when possible, to the form (i) is called the separation of the variables. For example, in the equation {\ — y)dx -\- {\-ir x)dy = o, (i) the variables are separated by dividing by (i — /)(i + ;tr) ; thus, -^ + ^:- = o (2) \ ->r X \ — y Hence, integrating, log(i + ^) - log(i -- y):=M c (3) 18. The integral here presents itself in a transcendental form ; but it is readily reduced to an algebraic form, for (3) may be written in the form log^-^ = ^; (4) whence 7^;=-- <=> or, putting C for ^, I + ^ = C(i - >') (6) It is to be noticed that C in equation (6) admits of all values positive and negative, although e^ can only be positive. In fact, equation (4) is defective in notation ; for, since the integrals are the logarithms of the numerical values of i + ;ir and \ — y respectively (see Int. Calc, Art. 10), that equation ought strictly to have been written and finally C is put for ±^. c. 1 6 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 1 9. 19. The complete integral in the above example is readily seen to be the equation of a system of straight lines passing through the point (—1, i). In general, any assumed simulta- neous values of x and y, that is, any assumed position of the moving point, determines a value of Cy as in Art. 13. But, for the particular point (—1, i), the value of C is indeterminate in equation (6) ; and accordingly we find that / or -^ is also inde- dx terminate for this point in equation (i). It must not, however, be assumed that, whenever / in the differential equation is indeterminate at a particular point, all the lines represented by the complete integral pass through the point in question. For, if the point be a node of the particular integral which passes through it, p will have, at this point, more than one value ; and, the differential equation being of the first degree, this can only happen when p takes the indeterminate form. For example, the integral of xdy + ydx = o is xy = C, representing a system of hyperbolas ; but the particular inte- gral which passes through the origin is the pair of straight lines xy =z o o( which the origin is a node. Accordingly / takes the indeterminate form at the origin. 'Ik Homogeneous Equations, 20. The differential equation of the first order and degree, Mdx -f Ndy = o, is said to be homogejieous when M and N are homogeneous functions of x and y of the same degree. Since the ratio of two homogeneous functions of the same degree is a function oiZ, a, homogeneous equation may be written in the X form .. 1 = 43 (■> §11.] HOMOGENEOUS EQUATIONS. y If, now, we put z/ for -, so that dy dv , y = vx, -^ = X-- + V, dx dx X "^^^ -- {^ " ^ the equation becomes '^ dv , ,/ s dx a differential equation between x and v in which the variables can be separated ; thus, dx dv X f{v) — V 21. For example, the equation {x^ — y^)~ — 2xy = o (i) dx is homogeneous. Putting ^ = vx, we obtain dv , 2V X h V dx 1 — v^ whence dv V + v^ X— = — ' , dx I — v^ or dx 1 — v^ y dv 2vdv dv = — X v{i + v^) V 1 + v^ Integrating, 1 8 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 21, y and, replacing t^ by -, or iog^i±J:! = c, x" -\- y^ = cy (2) 22. The geometrical meaning of the homogeneous equation (i) of Art. 20 is that -/ has the same value for all points at ax which ^ has a given value \ that is to say, if we draw a straight X line through the origin, the various curves of the system repre- sented have all the same direction at their points of intersection with this straight line. But this is the definition of similar and similarly situated curves, the origin being the centre of simili- tude. The curves of such a system are simply the different curves which would be constructed to represent the same equa- tion if we took different units of length. Denoting the unit of length by c, the general equation of the system will therefore be of the form (rf)-»- / where, since c is 'arbitrary, it is the parameter of the system ; in other words, the constant of integration c may be so taken that the complete integral of the homogeneous differential equation will be a homogeneous equation between x, 7, and c. In the example given in the preceding article, the system of curves represented consists of all circles touching the axis of x at the origin, and the final equation is so written that all of its terms are of the second degree with respect to x, y, and c. § IL] HOMOGENEOUS EQUATIONS. 1 9 Case in which the Functions M and N are of the First Degree. 23. The equation Mdx + Ndy = o can always be solved if M and N are functions of the first degree in x and y ; that is, when the equation is of the form dy ^ ax -\- by -\- c ^ , dx a'x + by -h c" ' ^ ^ for, substitute in this cC^- cL t. drj _ a^ + by] -\- ah -\- bk + c y = V + i,) ^.^c{y^ and we have ^^ x^d L ;i^ j_ ^7. _l_ AA _L >- ... (3) d$ a'$ + b'r] + a'h -\- b'k ■\- (f If, now, we determine h and k by the equations ah + bk -{- c = o, •:::! « a'h + <^'/& + c' equation (i) takes the homogeneous form dr] a^ -\- brj , from which we can determine the integral relation between i and t] ; and thence, by substitution from (2), the relation between x and y. 24. Equations (4) give impossible values of h and k when ^, b, a', and ^' are proportional. In this case, putting a' = ma, b' = 7nb, 20 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 24. equation (i) becomes dy __ ax -^ by -\- c dx m{ax + by) -f- /* Now put ax -\- by — z) whence dy _ \ dz a dx b dx b' Making these substitutions, we have dz , L z -^ c — =z a -h b dx mz -f- ^' an equation in which the variables can be separated. Examples II. Solve the following differential equations : — 1. (i 4- x)ydx + (i — y)^dy = 0, \ogxy = c — x -\- y. " "-^ T '^. 2. -^ = afXy ax^y + O' + 2 = o. dx 3. {y^ + xy^)dx + {x^ - yx')dy = o, log -^ = c, 4. xy{i H- x^)dy = (i + y^)dx, (i + ^^(i -f- jv^) = cx\ ^jc by — a dy f -\- 1 / . X §11.] EXAMPLES. 21 8. sin X cos y . dx — cos x^vsxy , dy, cos ^ = ^ cos x. Q. ax^ + 2y = xy^, x'^y'' = cey. dx dx ,o. ^ + ^ + ^^ + r ^ o, ^+>^+^ ^ c dx \ -\- X -\- x^ 2xy + X + y — 1 11. ^ + e^y = e^y^, log- ^ ~ ^ = ^-^ + ^. 12. ^^(I - /) = >'(! +^), log:>^_^~-^^ = ^. 13. xdy — ydx — sjipc" + f)dx = 0, jc^ = ^2 + 2cy, 14. (8); H- io.t)^^ + (5;^ + ^x)dy ■= o, {y + ^)^(.)' + 2xy = ^. 15- (^ + y)~- + ^ - >' = o, tan-^:^ + 41og(^2 + ^2) 3^ ^, dx X 16. (^>'-^0^ = ;'% ' ^=J. 17. ^ 4. ^^ = 2y, \Qg{x ^ y) ^ c - dx X — y (;; - ;t + \y{y + ^ - 1)5 = ^. 19. (:\:2 + f)dx — 2xydy = 0, jr^ — j^ = ex. 20. 2^jj;^jt: + (_);2 — TyX^^dy = 0, ^2 — jj;^ = ^jj;3. 21. y-" + (jcy + ^2)^ = 0, ^>^ = <^(^ -h 2j). 22. (^2 _ 2,y^^xdx + (3-^ — J>^)>'^ = o> 22 EQUATIONS OF FIRST ORDER AND DEGREE, [Art. 2$. III. Exact Differential Equations, 25. An exact differential containing two variables is an expression which may arise from the differentiation of a func- tion of X and y. Denoting the function by Uy we have du='^'±dx + ^dy, '. (I) ax ay where the coefficients of dx and dy are the partial derivatives of u. Thus, the form of an exact differential is Mdx + Ndy. But, if M and N are any given functions of x and y^ we cannot generally put du = Mdx + Ndy) (2) for, if ^ M = '^f, and N = '^, (3) dx dy we must have dM dN f . -di^-d^^ (4) d^u because each member of this equation is an expression for — — - . dxdy Hence equation (4) is a necessary condition of the possibility of equation (2) or equations (3) ; that is, of the existence of a function whose partial derivatives with respect to x and y are M and N respectively. 26. It is also a sufficient condition ; for the most general form of the function whose derivative with respect to x is My is '=( Mdx^Y, (5) where Mdx is integrated as if y were constant, and F is a quantity independent of x, but which may be a function of y. §111.] EXACT DIFFERENTIAL EQUATIONS. 23 Now the only other condition to be satisfied is that the derivative of u with respect to y shall equal N \ that is, dy] dy or ^ = iV^ - -^ f Mdx (6) dy dy] Since Y is to be a function of y only, but is otherwise unrestricted, this equation merely requires that the second member should be independent of x. This will be true if its derivative with respect to x vanishes ; that is to say, if dN__dM_^^ dx' dy This equation is iflentical with equation (4), which is, therefore, a sufficient, as well as a necessary, condition. 27. An equation in which an exact differential is equated to zero is called an exact differential equation. Using the notation of the preceding articles, the complete integral of the equation Mdx 4- Ndy =z o when exact is evidently u = C. The function u is determined by direct integration as indicated in equations (5) and (6). It is to be noticed that dV consists of those terms in Ndy which do not involve x ; and evidently the integral of these terms, and also of those containing x only, may be considered separately, and it is only necessary to ascertain whether the terms containing both x and y form an exact differential. For example, in the equation x(x 4- 2y)dx + (x"" — y^)dy = o. 24 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 2/. the sum of these terms is 2xydx -f x'^dy^ which is the differen- tial of x^y ; hence the complete integral is 1^3 + x-y - \y^ = C, or x^ H- z^y — y^ = c. 28. An expression involving only some function of x and y, and the differential of this function, is obviously an exact differential. Thus, in the equation xdx + ydy I ydx — xdy _ ^{x' -{- r - i) ~x^ + y^ ~ °' -^ the first term is a function of x^ + ^ and its differential, and the second is a function of - and its differential. The equation may, in fact, be written hence the integral is and, comparing this with equation (i) of Art. ^i, we see that — is an integrating factor of that equation. 30. A differential equation has a variety of integrating factors corresponding to different forms of the complete inte- gral. For example, one integrating factor of equation (i), 26 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 30. Art. 17, is the factor by means of which we separated the variables ; namely, I , (I +a:)(i - yY and this corresponds to the form (3) of the integral ; but, if we differentiate the integral in the form (5), Art. 18, wgl obtain equation (i) multiplied by the integrating factor (I - yy The forms of the integral differ in respect to the constants which they contain. In general, \i ti ^=- c is an integral giving the integrating factor /n, so that du — ^{Mdx + Ndy)y then f{u) = C where C = f{c) is another form of the integral ; and this gives the exact differential equation f{u)du = o, or f{u)iL{Mdx + Ndy) = o. Hence f{ti)\x. is also an integrating factor ; and, since /"denotes an arbitrary function, /' is also arbitrary ; thus, the number of integrating factors is unlimited. 31. The form of the given differential equation sometimes suggests an integrating factor. For example, in the equation (j + \ogx)dx — xdy — o, §111.] INTEGRATING FACTORS. 2/ the terms containing both x and y are ydx — xdy. This expression becomes an exact differential when divided either by y^ or by x^. The remaining term contains x only ; hence — is an integrating factor. Thus, we write ydx — xdy Xogxdx whence, integrating, —y log X I ^ X X ^ ' or <:^ + J/ -h log:v H- I = o. 32. The expression j/<3r;f — xdy, which occurs in the preced- ing article, is a special case of a more general one which should be noticed. For, since d{x^y^) = x^-'^y^-'^{mydx + nxdy), an expression of the form x^y^{mydx + nxdy) (i) has the integrating factor jQtn — i—ayn — i—P • and since, by Art. 30, the product of this by any function of //, where u = x'y*", is also an integrating factor, we have the more general expression ^km — x—a-ykn—r—^ ^2) (in which k may have any value) for an integrating factor. 28 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 32. As an illustration, take the equation yi^y^ 4- 2x^^dx -f x{jx^ — 2y^)dy 5= o. This may be written in the form y^(ydx — 2X£fy) + x*(2ytfx + xdy) = o, in which each term is of the form (i). In the first term, m = I, « = — 2, a = o, ^ = 3; and the expression (2) gives, for the integrating factor, that is to say, any multiplier of this form will convert the first term into an exact differential. In like manner, any expression of the form X2k'-5yk'-i is an integrating factor of the second term. A quantity which is at once of each of these forms will therefore be an integrating factor of the given equation. Equating the exponents of x^ and also the exponents of j, in the two expressions, we have — 2^ — 4 = k' — \j from which k ^ —2, and the integrating factor is x-^. Multiplying the given equation by ;t:-3, we have y^x-'^dx — 2y^x-^dy + 2xydx + x^dy = o; § III.] EXAMPLES. 29 and, integrating, or 20c^y — y^ — cx^. Examples III. Solve the following differential equations : — I. {pc^ — ^xy — 2y^)dx -\- i^f — a^xy — 2x'^)dy = o, xi -^ yi — dxyix -\- y^ z=z c. > 2. --^ = ^ ~ ^ , xy"^ = x^y 4- ^. rt!^ X 3jv^ — JC . 3. (2:^ — J + i)^:'c + {2y — X — i)dy = o, oc^ — xy •\- y^ •\- X — y =^ c. , 4. ^(:r2 4- 2>y^)dx + y{y^ + 3^)^ = o, X* + 6:r2>'2 -}. y^ = c. xdy — r^;c ^ + v^ y , 5. ^//j; + xdx + ^a ^ ^2 = o, — + tan-'- = c. V 6. (y — x)dy + ydx = o, logy -h - = c, V 7. ax-^y""-^ = 2^-=^ — y, ^ = -^ + ^. ' -^ ^:x: dx ^ n -h 2 X . 8. :v^ — y = ^(x^ — y^), sin-'=^ = logx -he, uX X 9. x^- y = xs/{x- + y), y = ^(^^- - -^V dx 2\ ce^ I 30 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 32. v/ 10, Q.^ — AT^)^ = o, x'y^ = jv^^c'* + ^. .12. (2jc2jj'' + y)t/x — {x^y — sx)dy = o, /[x^y = 5 + i:x^y^. ^3- (/* — 2jc3j)rt'jc + (;d — 2JC>'3)^ = 0, ^3 -j- ^3 = ^;cy. 14. Solve Exs. II., 19 and 20, by means of integrating factors. IV. T/ie Linear Equation of the First Order. 33. A differential equation is said to be linear when it is of the first degree with respect to y and its derivatives. The linear equation of the first order may therefore be written in the form ax where P and Q denote functions of x. Consider first the case in which the second member is zero, that is to say, the form ^ + ^ = (I) ax The variables may be separated ; thus, ^ = ^Mx, y § IV.] LINEAR EQUATION OF THE FIRST ORDER. 3 1 Hence or log J = ^ — \PdXf y=^Ce-\'" . (2) In this solution, Pdx may be taken to denote a particular integral of Pdx^ since the constant is directly expressed in the equation. 34. If we put equation (2) in the form and differentiate, we have eV'^^'^dy + Pydx) = o, which shows that e^^^ is an integrating factor of equation (i). Since Q is a. function of x only, it follows that e^ ^'^^ is also an integrating factor of the more general equation -f + iy = Q (') ax Hence, to solve this equation, we write e and, integrating, e ^^^y^^U^'^^Qdx + C, (2) In a given example, the integral? involved in the general expression (2) should, of course, be evaluated if possible. 32 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 34. Thus, let the given equation be (I +-)£= w + xy, ^. * V in dx I + ^'■' I +^ X tVif rpfrvrf* or Here P = jPdx= -ilog(i H-^0» and J^'^- = I is the integrating factor. Hence I X mdx and *'* + C, or J = WJt: + Csl{i + ^). Transformation of a Differential Equation. 35. It is frequently useful to transform a given differential equation by replacing one of the variables by a new variable which is an assumed function of the variable replaced, or of both variables. The form of the assumed function is generally § iV.] EXTENSION OF THE LINEAR EQUATION. 33 suggested by that of the given equation. Thus, the form of the equation (i -f xy)ydx + (i — xy)xdy = o suggests the use of a new variable v = xy^ whence xdy =. dv — ydx. Eliminating y, we have (i + v)-dx + (i — v)dv — (i — v)-dx = o, • X X or 2Tfdx + (i — v)xdv = o, in which the variables v and x can be separated. Integrating, 2 log .;c — - — log z; = ^, or and, substituting xj^ for ^, x = Cye^y, Extension of the Linear Equation. 36. If z; = f{y)i the linear equation for Vy becomes dx f{y)f^ + Ff{y) = C (0 In other words, an equation of this form becomes linear if we put V = f{y). 34 EQUATIONS OF FIRST ORDER AND DEGREE. [Art. 3/. For example, the equation ^+MX_(^_,)sec;- = o dx ^ + I takes the form (i) when multiplied by cos^, and hence is a linear equation for sin y. The integral will be found to be sin;; = ^^ - 3^ + ^ . 3(^ + I) . 37. In particular, the equation r^Ji/^ ^-^jy = Qr (2) is known as the extension of the linear equation. Dividing by y**, we have ax or (I _ n)y-n^ + (I - n)Py^— = (i - n)Q, ax which is of the form (i), and therefore linear for y^-^. Examples IV. Solve the following linear equations : — dv ^ \. -^ •\- y =■ Xf v = a:— 1+ ce-^, dx 2. -f = by + asmxy y = ce^'' — a '-^ . dx 1 -{- o^ ,3. ^ ^J— = {x + 1)3, 2y =^{x + lY + c{x + ly. ax X + 1 § ly.] EXAMPLES. 35 4. & ^ n^ =: ^^jf«, y = a;«(/ 7. -^ + y cos ^ = sin 2;*:, y = 2smx — 2 + ce- dx smjr 8. ^-^ — ^V = :x: + I, y = — + cx^. dx 1 — a a Y 9- -T^ + i^ 4. ^y = ^jx: I + ^ 2^(1 + x^y _ log [V^(i + ^') - i] - log^ + ^ •^ ~" 2sJ{l + ^^2) 10. ;v(l — ^2)_J: _|> (2a:2 — i)jv = ^:r3, jj; = ^^ -f cx\]{i — x^). II. cosx-^ + V — I + sinjv = o, y{sQCx + i3Xix) = x + c dx 13' (i + >'0^^ = (tan-*^ — x)dy, . ;«: = tan-^>' — I + r^-tan-'j- 36 EQUATIONS OF FIRST ORDER AND DEGREE, [Art. 37. Solve, by transformation, the following equations : — 14. -^ = w^ + «V + /, dx mnx + ify ■\- m •}- pn •= ce"^. y2 Y 15. {x — y)dx + 2xy£fy = 0, logx + -^ = c. 16. (x-h yy^ = a', ax = tan 17. ^ + >' = ^>'^ ;^ = ^ + i + ^^'" 18. -^ = ^31,3 ^ xy, -L =: ^2 _^ I + ^^-^'. fl'jc y^ 19/(1 - A^)-^ - ^j; = axf, ~ = rv/(i ~ x^) - a, dx y 20. 3/^ _ ^^3 = ^ + I, yl = ^^^ - ^^±JL _ -i dx a a 21. x^ + JJ' = /logjt:, - = log^ •\- \ -\- ex, dx y dx 2(1 — ^2) y 23. ^ = 1 , i = 2 _ y + ,,-*/. ' dx xy + X'y^ x § v.] DECOMPOSABLE EQUATIONS. 37 CHAPTER III. EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE. V. Decomposable Equations. 38. A differential equation of the first order is a relation between x, 7, and /. If the equation is not of the first degree with respect to /, the first step in the solution is usually to solve the equation for /. Suppose, in the first place, that the equation is a quadratic in / ; then two values of / in terms of X and y are found. These will generally be irrational functions of X and y ; in the exceptional case when they are rational functions, the equation will be decomposable into two equations of the first degree. For example, the equation may be written (l-X2-')=°- and is satisfied by putting dx X — o (2) or ^-y = <^ (3) ax 38 Equations not of the first degree. [Art. 38. The integrals of these equations are 2y = x^ -\- c (4) and y = Ce- (5) respectively. Each of these is therefore an integral of equa- tion (i). Thus, a decomposable equation of the second degree has two distinct solutions. Equations Properly of the Second Degree. 39. In a proper, that is, an indecomposable, equation of the /Second degree, the two expressions for / are the values of a two-valued function of x and y expressed by attaching the ambiguous sign to the radical involved. There is, in this case, 6ut one integral, the ambiguity disappearing in the process of mtegrating or of rationalizing the result ; so that it is, in fact, unnecessary to retain the ambiguous sign in the expression for /. For example, the equation gives / = ±^; whence fdyV ^ y \dxj X (0 dx , dy sjx sjy and, integrating, ^x ± ^y = ±\Jc (2) > But, rationalizing this, we have {x — yy - 2c{x + y) + c- — o, . . . . (3) a single equation for the complete integral. § v.] INDECOMPOSABLE EQUATIONS. 39 The system of curves represented by equation (3) consists of parabolas, each of which touches the two' axes at the same distance c from the origin, and the different combinations of signs in equation (2) simply correspond to the three arcs into which the parabola is separated by the points of contact. Systems of Curves corresponding to Equations of Different Degrees, 40. A differential equation of the first degree is, properly speaking, one in which / has a single value for given simulta- neous values of x and y. . An equation of the second degree is one in which / has two values for given values of x and j, and so on. Thus, such an equation as / = sin-'ji: is not an equation of the first degree, because, for any value of x, sin-';r has an unlimited number of values. The general form of an equation of the first degree is, then, Z/ + J/ = o, in which L and M denote one-valued functions of x and y. Two curves of the system cannot, in general, intersect, for in that case there would be two values of / at the point of intersection. The points, if there be any, at which L =: o and M = o, form an exception ; for, at these points, / is inde- terminate, as exemplified in Art. 19. Thus, the curves of the system either do not intersect at all, or intersect only at certain points where / is indeterminate.* It follows that, in the integral equation, given simultaneous values of x and y must, except in * The same reasoning shows that, the differential equation being of the first degree, points in which two arcs corresponding to the same value of c intersect, in other words, nodes, can only occur at the points where / is indeterminate. Con- versely, these points must either be points through which all the curves of the system pass, or else nodes. In the latter case, they may be crunodes through which two real arcs of a particular integral pass, or acnodes through which no arc passes. 40 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 40. the case of the points above mentioned, determine a single value of c, or, at least, values of c which determine a single curve. For example, the integral of the equation /= I +y (i) is Xzxi-^ y — X — o. (2) If we give particular values to x and 7, we find an unlimited number of values of a differing by multiples of tt ; but, writing the equation in the form y = tan(jf 4- a), we see that these values determine but a single curve. We, in fact, obtain all the curves of the system by allowing a to range in value from o to tt ; and, as a varies over this range, the curve sweeps over the whole plane once. If we take the tangent of each member of equation (2), and write tan a = ^, we have y — tan^ _ I + _ytan;t' in which any simultaneous values of x and y determine a single value of c ; and c must pass over the range of all real values in order to make the curve sweep once over the entire plane. 41. In general, if the constant of integration is such that different values of it always correspond to different curves, there can be but one value of c for each point ; hence the form of the integral is /{: 4- (2 = o V where P and Q are one-valued functions of x and y^ and this we may regard as the standard form of the integral. It will be noticed that both /* = o and Q = o sltq particular integrals ; § v.] SYSTEMS OF INTERSECTING CURVES. 4I the former corresponding to ^ = 00, and the latter to c =z o. Thus, in the example given above, j/ = tan x and jy = —cot x are particular integrals. 42. In like manner, the form of the differential equation of the first order and second degree is Lp' -^ Mp -{- N = o, where L, M, and N are one-valued functions of x and j/. In general, two curves, and two only, representing particular integrals, intersect in a given point. When the expression Z/2 _(- Mp + N can be separated into rational factors of the first degree, these curves belong to distinct systems having no connection with one another, as in Art. 38 ; but, in the general case, they are curves of the same system. Thus, the system of curves representing a proper equation of the second degree is a system of intersecting curves, two curves of the system, in general, passing through a given point. Hence, in the integral equation, given simultaneous values of x and j/ must generally determine two values of c, or, at least, values of c which deter- mine two and only two curves of the system. 43. Take, for example, the equation />' = ^ - y^ (i) or —-^ ^dx. ±v/(i - r) The integral is sin-'y — X = a, . (2) or y = sin(x -\- a), (3) in which it is permissible to drop the ambiguous sign, because j^ = — sin (^ -I- a) may be written j/ = sin (x -^ -n- -j- a), and is therefore included in the integral (3). Here, as in the example of Art. 40, if we give particular values to x and j/, a has an 42 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 43. unlimited number of values ; for, if 6 be the primary value of sin-' J, every expression included in either of the forms 2«7r + ^ or (2;z + i)7r — ^, where n is an integer, is a value of sin - ^x. These values of a, however, determine but two distinct curves, since values of a differing by a multiple of 2tt determine, in (3), the same curve, so that each of the above forms determines but one curve. Equation (3), in fact, represents the system formed by moving the curve of sines, y =. sin ;r, in the direction of the axis of x, and we obtain all the curves of the system while a varies from o to 27r, each branch or wave of the curve falling, when a = 27r, upon the original position of an adjacent branch. In this motion, the curve sweeps twice over that portion of the plane which lies between the straight lines y = i and j/ = —i, for which portion only the value of / is possible in equation (i). 44. If, in the integral of an equation of the second degree,, we so take the constant of integration c, that different values of it always correspond to different curves of the system, there can be but two values of c corresponding to a given point. The equation will then take the form Pc^ +Qc + R = o where P, Q, and R are one-valued functions of x and y ; and this may be regarded as the standard form of the integral. To reduce equation (3) of the preceding article to the stand- ard form, we have, on expanding, y = sin ^ cos a -}- cos :r sin a, in which sin a and cos a are to be expressed in terms of a single constant. For this purpose, we do not put sin a = r and cos a = v^(i — c^), because this would require us to introduce § v.] SINGULAR SOLUTIONS. 43 an irrelevant factor in rationalizing the equation in c ; but we express sin a and cos a by the rational expressions the sum of whose squares 'is unity; that is, we put \ — C^ 2C sin a = , cos a = I + (T^ I + r^ We thus obtain (c^^y + cos x) — 2r sin x + y — cos x = o, which is the complete integral of equation (i), Art. 43, ex- pressed in the standard form. Singular Solutions. 45. Representing a set of simultaneous values of x, y, and / by a moving point, every moving point which satisfies a given differential equation is, at each instant, moving in some one of the systems of curves representing the complete integral. In this sense, the latter completely corresponds to the differen- tial equation : nevertheless, there are, in some cases, other curves in which, if a point be moving, it will satisfy the differ- ential equation. For, suppose a curve to exist which, at each of its points, touches one of the curves representing particular integrals ; then a point moving in this curve is moving at each instant in the same direction, that is, with the same value of /, as if it were moving in a curve representing a particular inte- gral ; hence it satisfies the differential equation. Such a curve is an envelope of the system of curves repre- senting the complete integral, and its equation is called a singular solntion of the differential equation. An example has already been noticed in Art. 8 ; the equation xf' ^ yp -\- a =. o 44 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 45. has the singular solution in addition to the complete integral y ^ ex -ir-. c Now, the latter is the general equation of the tangents to the parabola y^ = 4ax, which accordingly form a system of straight lines of which the singular solution represents the envelope. 46. We shall now show how a singular solution, when it exists, may be found, either directly from the differential equa- tion, or from the complete integral if the latter be known. Two curves of the system touching the envelope at neighboring points intersect in a point which ultimately falls upon the envelope when the two curves are brought into coincidence ; hence the envelope is sometimes called the /ocus of the inter- section of consecutive curves. While the curves are distinct, two values of / in the differential equation correspond to the point of intersection. These become equal when the curves coincide, that is, when the point is moved up to the envelope ; and they become imaginary when the point crosses the envel- ope. In. like manner, if we substitute the coordinates of the point in the integral equation, it determines two values of c while the curves are distinct ; and these become equal when the point is moved up to the envelope, and imaginary when the point crosses it. Thus, at every point of the envelope, both the differential equation considered as an equation for /, and the integral equation considered as an equation for Cy have a pair of equal roots. Hence, if we form the condition for equal roots in either of these equations, which we shall, for shortness, call the /-equation and the r-equation, we shall have an equation which must be satisfied by every point on the envelope. § v.] THE DISCRIMINANT. 45 47. The expression which vanishes whenever two roots of an equation are equal is called the discriniijiant of the equation. The discriminants of the /-equation and the ^-equation are expressions involving x and y. Either of these expressions may break up into factors, the vanishing of any one of which causes the discriminant to vanish. Hence it follows from the preceding article that, if there be a singular solution, its equa- tion is the result of putting the /-discriminant, or one of its factors, equal to zero, and it is likewise the result of putting the ^-discriminant, or one of its factors, equal to zero. For example, in equation (i), Art. 43, the condition for equal roots is evidently y'^ — i — o ; hence 7—1 = and y -\- \ — o are the only equations which can possibly be singular solutions. Each of these equations gives, by differentiation, / := o, and is found to satisfy the differential equation. Hence they are singular solutions, the lines they represent being envelopes of the sinusoids represented by the complete integral. 48. The general method of finding the discriminant of an equation is to differentiate it with respect to the unknown quantity and then to eliminate that quantity between the result and the original equation. But, in the case of the equa- tion of the second degree, it is found more simply by solving the equation. Thus the /-equation, in this case, is Lf + Mp + N = o; (i) whence f=^K.±JiMlszAMl. (2) 2Z so that the condition for equal roots is M"^ - ^LN = o . (3) 46 EQUATIONS NOT OF THE FIRST DEGRJEE. [Art. 48. In like manner, the general form of the c-equation is and the condition for equal roots is Q" - A,PR == o. For example, in the final equation of Art. 44, the condition for equal roots is 4 sin^ X — 4 ( j^ — cos^ x) = o, or which is identical with the like condition for the /-equation given in Art. 47. Cusp-Loci. 49. There are other loci, for points upon which the discrimi- nants vanish, which it is necessary to distinguish from the envelope whose equation alone is a singular solution. There is, in fact, no reason why the values of p derived from the differential equation, when they become equal as the point (jT, y) crosses a certain locus, should also become equal to the value of / for a point moving along that locus. Suppose, then, that the two arcs' of particular integral curves passing through U', y) meet, without touching, the locus for which the values of p become equal ; and suppose, as will usually be the case, that the values of / become imaginary as we cross the locus ; then, when {x^ y) is moved up to the locus, the two arcs will come to have a common tangent ; and, since they cannot cross the locus, they will form a cusp, becoming branches of the same particular integral curve. Thus, the two values of c which § v.] CUSP-LOCI. 47 corresponded to the two intersecting arcs will also become identical, and the locus, which is called a cusp-locus, is one for which the ^-discriminant also vanishes. For example, the roots of the equation are equal, each being equal to zero, when ^ = o; but, since p =z co for a point moving along this line, this equa- tion does not satisfy the differential equation. The complete integral is {y + cy = x\ in which the condition of equal roots is x^ =■ o. The system of curves is that resulting from moving the semi-cubical par- abola j/^ = x^, which has a cusp at the origin, in the direction of the axis of y. This axis is, therefore, a cusp-locus. Tac-Loci and Node-Loci. 50. In the preceding article, the values of / were supposed to become imaginary as we cross the locus for which they become equal. From equation (2) of Art. 48, it appears that this will be the case if the discriminant changes sign,* but otherwise not ; hence, if the factor which vanishes at the locus appears in the /-discriminant with an even exponent, / will not become imaginary in crossing the locus. In this case, the two intersecting arcs cross the locus ; and, when {x, y) is moved up to the locus, we shall simply have two particular integrals which touch one another. Such a locus is called a tac-lociis. Since * Since the discriminant is the product of the squares of the differences of the roots, this will be true also for equations of the third and higher degrees. 48 EQUATIONS NOT OF THE FIRST DECREE. [Art. 50. the values of c for the two curves remain distinct, the factor indicating a tac-locus does not appear at all in the r-discrimi- nant, but appears in the /-discriminant with an even exponent.* 51. On the other hand, a factor may appear in the ^-discrimi- nant with an even exponent, and not at all in the /-discriminant. Through every point of the locus on which such a factor van- ishes, the proper number of arcs of particular integral curves pass, but two of them correspond to the same value of c ; thus, the point is a node of the curve determined by this value of Cy and the locus is called a node-loctis, 52. The equation xp^ — {x — ay — o (i) furnishes an example of each of the cases mentioned in the two preceding articles. The complete integral is y •\- c =^ \x^ — 2ax^, or Uy + cy = x{x- sciy (2) The /-discriminant is x(x - ay, (3) and the ^-discriminant is ' x{x - say (4) The system of curves is the result of moving the curve J^2 _ ^^^ _ 2^)2 * If a squared factor in the /-discriminant satisfies the differential equation, the two arcs of particular integral curves passing through (x, y), instead of crossing the locus when (jt, y) is moved up to it, will coincide with it in direction, as in the case of the envelope, Art. 46. But, since / is real on both sides of the locus, the arcs reappear upon the other side of the locus when {x, y) is moved across it. This implies that they coincide with the locus when {x, y) is upon it. Hence, in this case, the squared factor appears also in the ^-discriminant, and represents a particu- lar integral. § v.] TAC-LOCI AND NODE-LOCI. 49 in the direction of the axis of y. This curve touches the axis of y at the origin, has a node at the point (3^, o), and, between these points, consists of a loop in which the tangents at the two points where x ^=^ a are parallel to one another. Accord- ingly the factor x, which is common to both discriminants (3) and (4), indicates the envelope x =■ o\ x — a = o is a, tac- locus, and x — $a := o is a. node-locus. m 53. Two values of c become equal, in other words, the ^-dis- criminant vanishes, whenever the point (x, y) is at the ultimate intersection of consecutive curves of the system represented by the ^-equation. Suppose this equation to represent a curve having, for all values of c* one or more nodes or cusps. Considering the intersections of two neighboring curves of the system, it is evident that there are two intersections in the neighborhood of each node, and that these ultimately coincide with the node. Again, there are three (all of which may be real) which ultimately coincide with each cusp. Now, the ^-discrimi- nant gives the complete locus of the ultimate intersections : it therefore includes the node-locus repeated twice, and the cusp- locus repeated three times ; that is to say, the discriminant contains the factor indicating a node-locus as a squared factor, and it contains the factor indicating a cusp-locus as a cubed factor, as illustrated in the example of Art. 49, where the factor x^ occurs in the r-discriminant, while the first power only of x occurs in the /-discriminant. 54. A decomposable differential equation of the second degree has no singular solution : for the discriminant is the * If, for a particular value of c, a node occurs at the point {x, y), there are no intersections of consecutive curves in its neighborhood, the point does not cause the f-discriminant to vanish, and there are for it the proper number of values for r, and therefore one too many values of /. Hence, at such a point, the /-equation vanishes identically irrespective of the value of / ; that is to say, all its coefficients vanish. (See Cayley, Messenger of Mathematics, New Series, vol. ii. p. lo.) If a point cause both the /-equation and the ^-equation to vanish identically, it will be a fixed intersection of the curves of the system. so EQUATIONS NOT OF THE FIRST DEGREE. [Art. 54. square of the difference between the roots ; hence, if the roots are rational, it is the square of a rational function. The systems representing the two complete integrals, in this case, are non- intersecting systems ; and the discriminant vanishes only at the tac-locus, at every point of which a curve of one system touches a curve of the other system. Thus, in equation (i), Art. 38, the discriminant is {x -f yY - Axy - {x - yY, the square of a rational function ; and the line x =z y is 3. tac- locus at every point of which one of the parabolas represented by equation (4) touches one of the exponential curves repre- sented by equation (5).* Examples V. Solve the following equations, finding the singular solutions, when they exist, as well as the complete integrals : — ©■- Y = 0, y = ce^'^y and y = C^-«-^. 2. /(/ - 7) = x{x -f ;;), 2^ + ^ = ^i ^nd y -\- X + 1 = CV^. 3. {x^ -}- i)/2 = I, c^e^'y — 2cxey = i. * In like manner, the discriminant of a decomposable ^-equation gives a node- locus. But it is to be noticed that there is no propriety in combining the two integrals of a decomposable /-equation. Thus, if we combine equations (4) and (5) of Art. 38, assuming C and c to be identical, we associate each curve of one system with a particular curve of the other system. But if, before doing this, we change the form of one of the integrals (by introducing a new constant f{c), as explained in Art. 30), we associate the curves differently, and obtain a new result, equally entitled to be considered the integral of the given decomposable differential equation. § v.] EXAMPLES, 51 cev = x^'^. IdyV _a^ ^ ^ \dx) x^ 5. y-'f^ 4a^, y^ = c ± ^ax. 6- /"" — 5/ + 6 = o, y = 2X + c, and j' = 3;^ 4- C ^- (iy-f=°' (j'-.)==4-. 8. ^2^2 4- Z^yP + 2j'2 = 0, :ry = Cj and ^jc^^ = C 9. /3 _|- 2Ji:/2 y2p2 _ 2J|f>'2^ = o, y — ^} y + x^ = c, and x^' + i + ^j = o. 10. p^ — (xi^ + xy ■}• y^)/>^ + ^^^(^^ + xy + _y^)/ — x^y^ = o, rj' = e^^^, cy = I -\- xy, and 3jv = ^3 ^ ^, 11. /^ 4- 2/jvcot^ = y^j j(i ± cos^) = r. 12. ( — ) — ax^ = o, 2K(y — cY = 4ax^, \dxl 13. X 4- j(r/2 _ i^ y _ ^^^ _ ^2^ _|_ sin-'y/jc 4- c. 14. p\x- + 1)3 = I, (^ _ ^)^ = _^. x^ -\- \ 15. j; = (^ 4- i)/^, ^2 4- 2C{X 4- I 4- j^) 4- (^ 4- I _ ji;)2 = o. 16. jj//^ 4- 2^/ — ji; = o, j2 _ 2j:jt: ^ ^2, 17. 3^/2 — dyp 4- ^ 4- 2>' = o, <;2 ^ ^^^ _ 2^^ _l_ ^2 _ q^ 18. yp -ir nx=^ sj{y'^ + nx^)sj{i 4- p')j 52 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 54 19. oc^p^ — 2xyp -\' y^ — x^f -f ;d, 2C^ = c^e^^ — e-*, X 20. 3/*j^ — 2xyp 4- 4;;^ — :«:» = o, x^ ■\- y^ — A,cx 4- 3^' = o. 21. jv»(i + /») = «H^ + yp)\ {x -\- cy = {n" - i)y^ + «'^'. VI. Solution by Differentiation. 55. The differentiation of a differential equation of the first order gives rise to an equation of the second order ; but, in the cases now to be considered, the result may be regarded as an equation of the first order, and its integral used in determining that of the given equation. Let the given equation be solved for j/, that is to say, put in the form y = f{x,p)) (i) then the result of differentiation will be of the form which is of the second order as regards y, but, not containing y explicitly, is an equation of the first order between x and /. If, now, we can integrate this equation, we shall have a relation between x, /, and an arbitrary constant. The result of elimi- § VI.] SOLUTION BY DIFFERENTIATION. 53 _ nating / between this equation and equation (i) will therefore be a relation between x, y, and an arbitrary constant ; hence it will be the complete integral required. 56. For example, given the equation -^ ■\- 2xy = x^ -^ y^ \ ax solving for 7, we have y = x-^s^P', (1) and, differentiating, p = ^ + \f (2) 2^p ax Separating the variables x and /, we have dx = di_ and, integrating. 2>jp{p - I) ' or ^ ^^:^2^ ( ) Finally, eliminating / between equations (i) and (3), we have the complete integral C + e^^ y =2 X + C — e^^' 57. In attempting this mode of solution, it will sometimes be more advantageous to treat y as the independent variable, and dx putting /' for — -, to derive a differential equation involving y ay and /'. In either case, the success of the method depends upon our ability to integrate the derived equation. The princi- 54 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 57. pal cases in which this can be effected are those in which one of the variables is absent and those in which both variables occur only in the first degree. It should be noticed that the final elimination of p is frequently inconvenient, or even impracticable ; but, when this is the case, we may express x and y in terms of / which then serves as an auxiliary variable. Equations from which One of the Variables is Absent. 58. If an equation of the first order in which x does not occur explicitly can be solved for /, it takes the directly inte- grable form f = /w w y being treated as the independent variable. Otherwise let it be solved for y ; thus, J= (/); (2) differentiating, / = *'(/)£, C3) in which the variables x and / can be separated. In like manner, an equation not containing y^ if not directly integrable, should be put in the form Differentiating with respect to y, we have in which the variables y and / can be separated. § VI.] ONE VARIABLE ABSENT. 55 59. As an example, let us take the equation 7 = /^ + 1/3 (i) We have, by differentiation, / = (2/ + 2/^)^, (2) which implies either that / = o, (3) or else that dx ^ {2 -{■ 2p)dp (4) Eliminating / from equation (i) by means of the first of these, which is not a differential equation for/, we obtain the solution * 7 = o. (5) which does not contain an arbitrary constant. But, integrating equation (4), we have ^4-^=2/ + /% or / = — I -f ^{x -f c)\ and, employing this result to eliminate / from equation (i), we obtain or, rationalizing, {x^- y ^- c-\y = %{x^ cy (6) This equation contains an arbitrary constant, and is the com- plete integral. Equation (5), not being a particular case of equation (6), is a singular solution. $6 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 6o. 60. With respect to an equation of the form >' = (/), (i) y = <^(o), (2) it may be noticed that (which, since <^ is not necessarily one-valued, may include several equations) is always a solution, for it gives, by differ- entiation, / = o, and thus satisfies equation (i). The reason of this is readily seen, for the complete integral is capable of expression in the form X = ^{y) -\r c, (3) which is the form it would take if derived by direct integration from the form (i). Art. 58 ; it therefore represents the system of curves which results from moving the curve in the direction of the axis of x. If this curve contains points at which / = o, it is evident that the locus of these points, or y = (o) will be the particular integral corresponding to ^ = 00 when the integral is written in the form (3). For * If the /-discriminant were formed, in this case, by the general method (see Art. 48), we should apparently have satisfied by j = o. This is, of course, not a singular solution ; but the complete integral is log J z^ X -\- c or y ^=z Ce^, and j/ = o is the particular integral corresponding to ^ = — 00 in the first form, or to 6" = o in the second. Homogeneous Equations. 61. When a homogeneous equation which is not of the first degree can be solved for /, it takes the form dx \x) considered in Art. 20. Otherwise it should be put in the form or y = x{p) (i) Differentiating, P = HP) + ^*'(/)f » (2) in which the variables can be separated. 62. If /i is a root of the equation / = <^(/), y = p^x is always a solution of equation (i) ; for it gives, by differentia- tion, / = /„ and substituting these values in equation (i), we have p,x = X(f>{p,), which is satisfied by the hypothesis. It was shown in Art. 22 that the complete integral, in this case, represents a system of similar curves with the origin as 58 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 62. the centre of similitude. It is hence evident that the tangents from the origin to any curve of the system will, if the points of contact be at a finite distance, constitute the envelope of the system ; but, if the points of contact be at an infinite distance, they will be asymptotes to the system. In either case, they y will be the loci of the points for which / = - in the differen- tial equation (i), that is to say, for which p — <^(/) ; but, in the first case, their equations will be singular solutions ; * and, in the second case, they will constitute the particular integral corresponding to ^ = o when the complete integral is written in the homogeneous form, as in Art. 22. Equation of the First Degree in x and y. 63. The equation of the first degree in x and y may be written in the form y = x{p) + f{p) (i) Differentiating, we have /=,/.(/) +^.^'(/)2 + /'(/)£,. ... (2) or dp p- (/>(/) ^ /-(/)' . . . . u; which is a linear equation for x regarded as a function of /. The integral gives ;ir as a function of / ; the elimination of / is often impracticable, but, in that case, substituting the value of X in equation (i), we have x and y expressed in terms of / as an auxiliary variable. * In this case also, <^\p) — o determines cusp-loci, but fails to detect a tac- locus. See the preceding foot-note. § VI.] CLAIRAUrS EQUATION. 59 Clairaufs Equation, 64. The equation y = px^f{p), ; (i) which is a special case of equation (i) of the preceding article, is known as Clairaut's equation. The result of differentiation ^ = ^ + .| + /(,)|. or "' '^ o. [. + r(/)]| This equation is satisfied either by putting ^ + /(/) = 0, (3) or by putting ^ = o. (4) ax Equation (3) gives, by the elimination of / from (i), a singular solution ; and equation (4) gives, by integration, whence, from (i), y ^ ex -^ f{c). (5) This is the complete integral, as is verified at sight, since / = ^ is the result of its differentiation. 65. The complete integral, in this case, represents a system of straight lines, and the singular solution a curve to which these lines are tangent. An example has already been noticed in Art. 45. Conversely, every system of straight lines repre- 6o EQUATIONS NOT OF THE FIRST DEGREE. [Art. 65. sented by a general equation containing one arbitrary parameter gives rise to a differential equation in Clairaut's form, having, for its singular solution, the equation of the curve to which the system is tangent. We have only to write the equation in the form (5), and to substitute / for the symbol denoting the param- eter. For example, the equation of the tangents to the circle x' -\- y^ = a?- is y = mx 4- «V(i + ^«'); hence the differential equation is y^ px^ aslii -f /^); or, rationalizing, {x^ — a^)p^ — 2xyp + >'2 — a= = o. Accordingly the condition of equal roots is found to be xy — (x^ — a"") (y — a^) = o, or x"" -]- j/^ = a^. 66. If we form the condition for equal roots in equation (i). Art. 64, by the general method mentioned in Art. 48, we have to eliminate p from equation (i) by means of its derivative with respect to / ; namely, o = x±f\p), tft] which is identical with equatlBfc). In fact, it is obvious that the condition should be the^Rie ; for, since the complete integral represents straight lines, there can be neither cusp- locus nor tac-locus. Precisely the same condition expresses the equality of roots in the ^-equation, a node-locus being also impossible. § VI.] REDUCTION TO CLAIRAUT'S FORM. 6l 67. A differential equation may be reducible to Clairaut's form by a more or less obvious transformation. For example, given the equation j,_,^| + «^gJ = o; since d{y^) = ^ydy, if we multiply through by y, y'^ may be made the dependent variable ; thus, f-,^l§L + aim = o, ax \ ax I or, putting y'^ = v^ hence the integral is dx ^\dx) y^ = ex — lac^. c Examples VI. Solve the following differential equations : — I. ^ = —xp + x^p^, y =z - -\- c^ X singular solution, i + ^x'^y — o. 2. xp^ — 2yp -{- ax = o, 2y — cx^ -\-^ . c 9 y = T- singular solution, y'' = ax- S. X -{- py{2p' + 3) = o, c ^ cp{2p- + 3) (I + /O^' ^ (I + p')^ 4. y = ^^P' X ^ ""^^ - ^^ + C 62 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 67. 5. .Y + J// = tf/», ^= ^(x :^^^ i^-^^^Qg[/ + v/(i ^-/^)]^ 6. ;' = (I + /)^ + /^, (^ = 2(1 — /) + ^^-i*, b = 2 - /^ + r^->(i + /). X = a\og [ay + yl{a' + J^ — 1)] 4- \og\_y - sl{a^ + J* - i)] 4- ^. 8. 2>' = ;r/ + 2> a^c^ — i2acxy + ^cy"^ — \2x^f- + i6^^3 = o. 9. ^ = «/ + ^/2, X = a\ogl\J{a^ + 4/^;^) - «] + v/(«' + 4^J') + ^. 10. «'_);/* — 4jc^ + ^ := o, 11. ^ = ^/ + V(3^ + «^/^), ^ = ^^ + v/(^^ + «V^), singular solution, }- ^ = i . 12. (l + ^0/^ — 2^_)^/ 4- j;2 _ J _ Q^ jj, _ ^;^; ^ y/(l _ ^2). 13. y = ^(jc — <5) 4- -, singular solution, y^ = 4<3;(a: — ^). 14. «^;/>^ 4- (2^ — d)J> — y = o, ac^ + c{2x — d) — y = o. V 15. fi -^" = e-y + e—-(£\\ ^y = ce^ + sj{i -\- c^). 16. a:2(;; - px) = :>;/*, ^2 = ^^ 4- c\ 17. ' = 0, eY z= ce^ ^ c^^ § VIL] EXAMPLES. 63 18. {af- — b)xy -\- {bx" — ay"- + c)p = o, Cc f ^Cx" -^r b ^ aC '9- f-^ = 4|-f^)' y^c^^^m. 20. /3 _ ^xyp + 8y2 = o, y = c(x — cy. 21. x'^p^ — 2{xy — 2)p + y^ s= o^ {y — ^^Y + 4^ = o. 22. y •= 2px + y'^p^j y^ z= ex -\- J^3. 23. (/^ — y) {py + x) == h^py f ^ CSC" ^ — c + I VII. Geometrical Applications. 68. The properties of a curve are frequently expressed by means of such magnitudes as the subtangent, the subnormal, the perpendicular from the origin upon the tangent, etc., the general expressions for which involve the coordinates of a point upon the curve together with the value of the derivative at that point. Hence the analytical expression of such a property, or, indeed, of any property which depends upon the tangents to the curve, gives rise to a differential equation. Again, a property relating to an area or volume connected with a curve, or to the length of an arc of the curve, is expressed by a differential equation. Hence the problem to determine the curve having a given property resolves itself into the solution 64 GEOMETRICAL APPLICATIONS. [Art. 68. of a differential equation. For example, the expression for the subnormal is yp ; hence, to determine the curve whose sub- normal is constant and equal to a, we have only to solve the differential equation dy The integral of this equation is therefore the curve having the given property is the parabola wlttjee parameter is 2a, and whose axis is the axis of ;r, the position of the vertex being indeterminate. 69. The given property is, in some cases, expressed in polar coordinates. Thus, let it be required to determine the curve in which the angle between the radius-vector and the tangent is 7t times the vectorial angle. Using the expression for the trigonometric tangent of the angle first mentioned, the prop- erty is expressed by the equation — = ia.nn6, dr or dr cos nQdB Integrating, r sin nB log r == - log sin nO + C, which may be written in the form The mode in which the constant enters shows, as might have been anticipated, that the several curves which have the prop- erty are simply similar curves similarly situated with respect to the pole ; thus, when n = i, they are the circles which touch the initial line at the pole. § VIL] POLAR COORDINATES. 65 70. As a further illustration, let us consider the curve traced by a point carried by a curve which rolls upon a fixed straight line. By the principle of the instantaneous centre, the straight line joining the carried point with the point of contact of the curve with the fixed line is always normal to the path of the carried point. Considering the carried point as a pole, this line is a radius-vector of the given curve, and the perpendicular from the carried point to the fixed line is the perpendicular from the pole upon a tangent. Denoting these lines by r^ and /, respectively, the nature of the given curve determines a relation between r, and /,. But, taking the fixed line as the axis of x, /, is an ordinate of the required curve, and r, is the part of the normal intercepted between the point of contact and the axis of x, the expression for which is ysj^i + p^). The relation between /, and r, then at once gives the differential equation. For example, let the parabola y^ = d^ax roll upon a straight line, and let it be required to determine the curve traced by the focus. The relation between p^ and r„ in this case, is therefore the differential equation is f = aysl{i + f), or, solving for /, dy _ yj{y^ — a^) dx a Let us take as the origin the point of the fixed line on which the vertex of the parabola falls in the rolling motion. This deter mines the constant of integration by the condition that ^ = o when ^ = o, that is to say, when y = a. Integrating, we have f ^y = f ^ 66 GEOMETRICAL APPLICATIONS. [Art. 70. or y + ^lif-a^) = f . ^ a a which may be reduced to the form _y = ^M + 1?"^ = a cosh-. The curve is the catenary. 71. In another class of examples, the curve required is the singular solution of a differential equation. It is, in this case, frequently possible to write the complete integral at once, and to derive the singular solution from it instead of forming the ydifferential equation. For example, required the curve such that the sum of the intercepts of its tangents upon the axes is constant and equal to a. The equation of the curve is the singular solution of the equation whose complete integral represents the system of lines having the property mentioned. The general equation of this system is X y 7 + -r=r-c = ^> in which c is the arbitrary parameter. Writing it in the form c'^ + c{y ^ X — a) -{- ax = o, the condition of equal roots is (y — X — ay — ^ax = o, or (y — xY — 2a{x -\- y) -\- a^ = o, which is the equation of the required curve, and represents a parabola touching the axes at the points (a, o) and (o, a). § VII.] TRAJECTORIES. 6^ Trajectories. 72. A curve which cuts a system of curves at a constant angle is called a trajectory of the system. The case usually considered is that of the orthogonal trajectory, which cuts the system of curves at right angles. The differential equation of the trajectory is readily derived from that of the given system of curves ; for, at every point of the trajectory, the value of p has a fixed relation to the value of / corresponding to the same values of x and y in the equation of the given system of curves. Denoting the new value of / by /', this relation is, in the case of the orthogonal trajectories, '=-?■ If, then, we put in place of ^ in the differential equa- dy dx tion of the given system, the result will be the differential equation of the trajectory. The complete integral of this equa- tion will represent a system of curves, each of which is an orthogonal trajectory of the given system. Reciprocally, the curves of the given system are the orthogonal trajectories of the new system. 73. For example, let it be required to determine the orthog- onal trajectories of the circles which pass through two given points. • Taking the straight line which passes through the two given points as the axis of y and the middle point as the origin, and denoting the distance between the points by 2^, the equation of the given system of circles is oc^ -\- y'^ -\- ex — b^ =i Q, (i) in which c is the arbitrary parameter. The differential equation derived from this primitive is (x^ — y^ -\- b'^)dx + 2xydy = o (2) 68 GEOMETRICAL APPLICATIONS. [Art. yi. Substituting for -/, we have ay ax (^y^ — ^' — b^)dy + 2xydx = o . . . . (3) for the differential equation of the trajectories. This equation is the same as the result of interchanging x and y in equa- tion (2), except that the sign of b^ is changed ; its integral is therefore x^ ^ y^ -^Cy -^ b^ = o; (4) and the trajectories form a system of circles having the axis of X as the common radical axis, but intersecting it and each other in imaginary points. 74. It is evident that the differential equations of the given system and of the orthogonal trajectories will always be of the same degree, and that, wherever two values of / become equal in the former, the corresponding values of / will be equal in the latter. Hence the loci of equal roots will be the same in each case. Now, the trajectories will meet an envelope of the given system at right angles ; and, since the values of / become imaginary in both equations as we cross the envelbpe, the envelope is a cusp-locus of the system of trajectories. Conversely, a cusp-locus which is, at each point, perpendicular to a curve of the given system, becomes an envelope of the system of trajectories ; but every other cusp-locus is also a cusp-locus of the trajectories. In like manner, a tac-locus of the given system becomes a tac-locus of the trajectories.* A node-locus gives rise to no peculiarity in the system of trajectories. * The case in which the tangent curves of the system cross the tac-locus at right angles forms an exception. In this case, the locus is itself one of the trajectories ; and being represented, in the common /-discriminant of the two systems, by a squared factor, we have the case considered in the foot-note on § VII.] EXAMPLES. 6g Examples VII. 1. Determine the curve whose subtangent is n times the abscissa of the point of contact. jj;« = ex. 2. Determine the curve whose subtangent is constant, and equal to a. ce^ = y^. 3. Determine the curve in which the angle between the radius- vector and the tangent is one-half the vectorial angle, r = ^(i — cos^). 4. Determine the curve in which the subnormal is proportional to the n\h power of the abscissa. y^ = >^.r«+ ' + c. 5. Determine the curve in which the perpendicular upon the tangent from the foot of the ordinate of the point of contact is constant and equal to a, determining the constant of integration in such a manner that the curve shall cut the axis of y at right angles. The catenary y = a cosh-. a page 48. For example, the tac-locus x z=:am Art. 52 is perpendicular to the system of curves representing the complete integral ; the equation of the trajectories is (x — aYp^ — X = 0,. (i) of which the integral is y + C = 2^x + \Ja\ogi^^-^^ (2) \a + Sx The system is that which results from moving the curve in the direction of the axis of y. This curve is symmetrical to the axis of x since Sx admits of a change of sign, and it has a cusp at the origin, so that the axis of y is a cusp-locus. The line x z= a'xs an asymptote which is approached by branches on both sides of it ; and the result of putting C = 00 in equation (2) is, in fact, this line, or rather the line doubled, for, if C is infinite, we must, in order to have y finite, put x z=. a. yo GEOMETRICAL APPLICATIONS. [Art. 74. 6. Determine the curvef in which the perpendicular from the origin upon the tangent is equal to the abscissa of the point of contact. X^ •\- f — 2CX. 7. Determine the curve such that the area included between the curve, the axis of x^ and an ordinate, is proportional to the ordinate. 8. Determine the curve in which the portion of the axis of x intercepted between the tangent and the normal is constant, and interpret the condition of equal roots for /. 2{x - c) = a\o^{a ± sl{a^ ~ 4^^')] T yj{a^ - 4/)- 9. Determine the curve such that the area between the curve, the axis of X and two ordinates is proportional to the corresponding arc. y = cosh X 10. Determine the curve in which the part of the tangent inter- cepted by the axes is constant. x^ -^ yi — ah 11. Determine the curve in which a and yS being the intercepts upon the axes made by the tangent ?na -f- n/S is constant. The parabola {ny — fnx)^ — 2a{ny + mx) ■\- a^ — o. 12. Determine the curve in which the area enclosed between the tangent and the coordinate axes is equal to a^. The hyperbola 2xy = a^. 13. Determine the curve in which the projection upon the axis of y of the perpendicular from the origin upon a tangent is constant, and equal to a. The parabola x^ = 4^(^ — y). 14. Determine the curve in which the abscissa is proportional to the square of the arc measured from the origin. The cycloid v — asin~'^ -f yj{ax — x^). 15. Determine the orthogonal trajectories of the hyperbolas xy = a. The hyperbolas x^ — y^ — c § VII.] EXAMPLES. 71 16. Determine the orthogonal trajectories of the parabolas y = 4^^. The ellipses 2x^ -{- y^ = c^. 1 7. Determine the orthogonal trajectories of the parabolas of the ;zth degree a«-'jv = x''. ny'^ + x^ = c"". 18. Find the orthogonal trajectories of the confocal and coaxial parabolas y^ = 4a{x + a). The system is self-orthogonal. 19. Show generally that a system of confocal conies is self- orthogonal. 20. Find the orthogonal trajectories of the ellipses — + -^ = i when a is constant and d arbitrary. x^ -^ y^ = 2a^ log x -^ c. 2 1 . Find the orthogonal trajectories of the cardioids r = ^ ( i — cos 6) . r = c{i -\- cosO). 22. Determine the orthogonal trajectories of the similar ellipses ~ -{- ^ = ?i^, n being the arbitrary parameter. y^^ = cx^^. 23. Find the orthogonal trajectories of the ellipses —^-\-^^—\ when- + - = 1. {xyY^ = ^^^'+>'^ 24. Find the orthogonal trajectories of the system of curves 25. Find the orthogonal trajectories of the curves ;- = log tan Q -\- a. - = sin^'^ + c. r 72 EQUATIONS OF THE SECOND ORDER, [Art. 75. CHAPTER IV. EQUATIONS OF THE SECOND ORDER. VIII. Successive Integration. 75. We have seen, in Chapter I., that the complete integral of a differential equation of the second order must contain two arbitrary constants, and that it is the primitive from which the given differential equation might have been derived by differentiating twice and using the results to eliminate the constants. The order in which the differentiations and elimi- nations take place is evidently immaterial ; for, denoting the constants by r, and c^^ and the first and second derivatives of jv by / and ^, all the equations which can arise in the process form a consistent system of relations between x^ j/, c^, C2, /, and qy and these are equivalent to three independent algebraic relations between these six quantities. If, after differentiating the primitive, we eliminate the constant c^, the result will be a relation between Xj y, c„ and /, that is to say, a differential equation of the first order ; and, if we further differentiate this equation, and eliminate r„ the result will be the differ- ential equation of the second order. Now, regarding the latter as given, the relation between x, y, r„ and / is called a first integral \ and the complete integral, or relation between x^ y^ c,y and ^2, is also the complete integral of this first integral, c^ being the constant introduced by the second integration. § VIIL] SUCCESSIVE INTEGRATION. 73 76. As an illustration, let the given equation be If this be multiplied by 2/, it becomes ^^1+^^!=°' « and, since this equation is the result of differentiating f ^- y^ = c^ ^ \ (3) (the constant, which is, for convenience, denoted by c^, dis- appearing in the differentiation), equation (3) is a first integral of equation (i). It may be written and its integral, which is sin-'— = :^ -f a, or y = c^m{x -^ a), (4) where a is a second constant of integration, is the complete integral of equation (i). Expanding sin {x + a), and putting I A = ^COSa, B = ^sina, the complete integral may also be written in the form y = Asinx ■}- Bcosx, (5) in which A and B are the two arbitrary constants. 74 EQUATIONS OF THE SECOND ORDER. [Art. 7/. The First Integrals. 77. It is shown, in Arts. 14 and 15, that a differential equation of the second order represents a doubly infinite system of curves. In fact, if, in the complete integral, we attribute a fixed value to one of the constants, we have a singly infinite system ; and, therefore, corresponding to different values of this constant, we have an unlimited number of such systems. For example, if, in the complete integral (4) of the preceding article, we regard f as a fixed constant, the equation represents a system of equal sinusoids each having the axis of X for its axis and c for the value of its maximum ordinate, but having points of intersection with the axis depending upon the arbitrary constant a. The first integral (3) is the differ- ential equation of this system ; and equation (i), which does not contain ^, represents all such systems obtained by varying the value of c. On the other hand, if, in equation (4), we regard a as fixed, we have a system of sinusoids cutting the axis in fixed points, but having maximunj ordinates depending upon the constant c, which is now regarded as arbitrary. If now we differentiate equation (4) and eliminate c^ we have the differential equation of this system, namely, 7 = /tan(:c -f a), (6) which, being a relation between x, j/, / and a constant, is another first integral of equation (i). The result of eliminating / between the first integrals (3) and (6) would, of course, be the complete integral (4). 78. Consider now the form (5) of the complete integral. If we regard A as fixed, the singly infinite system represented is one selected in still another manner from the doubly infinite system ; it consists, in fact, of those members of the doubly infinite system which pass through the point (Jtt, A). The § VIIL] THE FIRST INTEGRALS. 75 differential equation of this system, which is found by differen- tiating, and eliminating B, is jsinjx: -f pco^x = Af (7) which is, accordingly, another first integral of equation (i) Again, regarding B as fixed, and eliminating A from equation (5), we obtain the first integral ycosx — psinx = B (8) In like manner, to every constant which may be employed as a parameter in expressing the general equation of the doubly infinite system of curves there corresponds a first integral of the differential equation of the second order. Thus, the number of first integrals is unlimited. 79. If c, and C2 are two independent parameters, that is to say, such that one cannot be expressed in terms of the other, all the other parameters may be expressed in terms of these two. Accordingly, the two first integrals which correspond to c^ and <:2, which may be put in the form /i(-^> y^ P) = ^i, fzk^y y, P) = ^2» may be regarded as two independe^it first integrals from which all the first integrals may be derived. For example, if the first integrals (7) and (8) of the preceding article be regarded as the two independent first integrals, equation (3) of Art, 'j6 may be derived from them by squaring and adding, because c^ = A"- -\- B\ It must be remembered that no two first integrals are independent when regarded as differential equations of the first order ; for they must both give rise, by differentiation, to the same equation of the second order. They are only inde- pendent in the sense that the constants involved are independ- ent, so that they may be regarded as independent algebraic 76 EQUATIONS OF THE SECOND ORDER. [Art. 79. relations between the five quantities x^ j, /, ^„ and c^^ from which, by the elimination of /, the relation between x, j, f„ and fj can be found independently of the differential relation between x^ y^ and p. Integrating Factors. 80. If a first integral of a given differential equation of the second order be put in the form f{xy y^ p) z=z c and differen- tiated, the result, not containing c^ will be a relation between X, y, py and ^, which is satisfied by every set of simultaneous values of these quantities which satisfies the given differential equation. This result will therefore either be the given equa- tion, or else the product of that equation by a factor which does not contain {/. In the first case, the given equation is said to be an exact differential equation ; in the latter, the factor which makes it exact is called an integrating factor. In general, to every first integral there corresponds an integrating factor. For example, differentiating equations (7) and (8) of Art. 'j'^y we find the corresponding integrating factors of the equation doc^ to be cos X and sin x respectively. Again, the integrating factor / was employed, in Art. ^6, in finding the first integral (3) by means of which we solved the equation. 81. It is to be noticed that an exact equation formed, as in the case last mentioned, by means of an integrating factor containing /, is really a decomposable equation consisting of the given differential equation of the second order and the differencial equation of the first order which results from putting the integrating factor equal to zero. The exact differ- ential equation therefore represents, in this case, not only the doubly infinite system, but also a singly infinite system which does not satisfy the given differential equation. This system § VIII.] INTEGRATING FACTORS. J J consists of the singular solutions of the several singly infinite systems represented by the first integral when different values are given to the constant contained in it. For example, equa- tion (2), Art. j6, is satisfied by j/ = (7, which does not satisfy equation (i), but is the solution of / = o; accordingly, the first integral (3) has the singular solutions y =: ±c^ which, when c is arbitrary, form the singly infinite system of straight lines parallel to the axis of x. In fact, a singular solution of a first integral represents a line, which, at each of its points, touches a particular curve of the doubly infinite system. The values of X, f, and /, for a point moving in such a line, are therefore the same as for a point moving in a particular integral curve ; but the values of q are, in general, different ; * hence such a point does not satisfy the given differential equation. * The values of ^ will, however, be the same if the line in question has at every point the same curvature as the particular integral curve which it touches at that point ; and its equation will then be a singular solution. The case is analogous to that of the singular solution of an equation of the first order ; the given equation being supposed of a degree higher than the first in ^, and a necessary (but not a sufficient) condition being that two values of ^ shall become equal for the values of x, y, and / in question. Suppose, for example, the doubly infinite system of curves represented by the differential equation to consist of all the circles whose centres lie upon a fixed curve. In order to determine the particular integrals which pass through an assumed point [x, y) in the direction determined by an assumed value of /, we must draw a straight line through [x, y) perpendicular to the assumed direction, the required particular integrals being circles whose centres are the points where this line cuts the fixed curve. These circles correspond to the several values of q which are consistent with the assumed values of x, y, and /. When the line touches the fixed curve, two of the values of q are equal, and the values of x, y, and / satisfy the condition of equal roots in the differential equation considered as an equation for q. Consider now an involute of the fixed curve ; its normals touch the given curve ; hence the values of x, y, and /, at any of its points, satisfy the condition of equal roots. Now, the circle corresponding to the twofold value of q is the circle of curvature of the involute, so that the value of q for a point moving in the involute is the same as its value for a point moving in a particular integral curve, and the equation of the involute is a singular solution. Thus the involutes of the fixed curve constitute a singly infinite system of singular solutions, and the relation between x, y, and /, which is satisfied yS EQUATIONS OF THE SECOND ORDER. [Art. ^2. Derivation of the Complete Integral from Two First Integrals. 82. It sometimes happens that it is easier to obtain two independent first integrals than to effect the integration of one of the first integrals. The elimination of / between the two first integrals then gives the complete integral. For example, as an obvious extension of the results obtained in Art. 80, we see that both cos ax and sin ax are integrating factors of the equation and, since these expressions contain x only, they are also integrating factors of the more general equation;- + '^^^ = ^ w if X is a function of x only. Thus, we have the exact differ- ential equation, CQsax—^ + a^yQ,0'?>ax — X 0,0% ax. dx^ ^ and its integral, which is cos^jc--^ -f- ay€vciax = \X co'?> axdx ■\- c^ , . . (2) dx J is a first integral of equation (i). In like manner, the integrat- ing factor sin ax leads to the first integral ^vaax-^ — ^^cos^;i; = X sin ^^if^jc — c^. , . . (3) dx J by all the involutes (in other words, their differential equation) satisfies the con- dition of equal roots; that is to say, it is the result of equating to zero the discriminant of the ^-equation or one of its factors. § VIIL] ELIMINATION OF p FROM TWO FIRST INTEGRALS. 79 Eliminating / between equations (2) and (3), we have ay = sin ax X cos axdx — cos ax X sin axdx + c^ sin ax + c^ cos ax^ the complete integral of equation (i). 83. The principle of this method has already been applied to the solution of equations of the first order in Art. 55. The method there explained, in fact, consists in forming the equa- tion of the second order of which the given equation is a first integral, then finding an independent first integral, and deriving the complete integral by the elimination of /. But it is to be noticed that the given equation, containing, as it does, no arbi- trary constant, is only a particular case of the first integral of the equation of the second order corresponding to a particular value of the constant which should be contained in it. Accord- ingly, the final equation is the result of giving the same par- ticular value to this constant in the complete integral of the equation of the second order. For example, in the solution of Clairaut's equation. Art. 64, the equation of the second order is —^ = o ; the first integral, of which the given equation is a special case, is jj/ -f- C ^=^ xp -\- f{p) ; and the complete inte- gral \^ y -\- C = ex -^ /(c), which represents all straight lines ; whereas the required result is the singly infinite system of straight lines corresponding to 6' = o.* * In accordance with Art. 81, it would seem that a singular solution of the given equation, when it exists, could not satisfy the equation of the second order, and therefore must correspond to a factor which divides out, just as x -\- f'[p) does in the solution of Clairaut's equation. This is indeed true when the singular solution belongs to the generalized first integral, as in this case it does to jj/ + C =.cx 4-/(;2 — ^2(1 — a:^) = o, . . . . (5) which represents a system of conies having t^eir centres at the origin, and touching the straight lines x ■= ±1. 83 EQUATIONS OF THE SECOND ORDER. [Art. ^6, Equations in which y does not occur, 86. A differential equation of the «th order which does not contain y is equivalent to an equation of the (« — i)th order for/. The value of / as a function of x obtained by integrating this will contain n — i constants ; and the remaining constant will appear in the final integration, which will take the form y =» \pdx 4- C, If the given equation is of the first degree with respect to the derivatives, it will be a linear equation because the coefficients do not contain y. Thus, if the equation is of the second order, it may be put in the form gn- /(.)! = ,(.), or a linear equation of the first order for /. For example, the equation dx^ dx is equivalent to dx I + x^^ I + ^' The integral of this is / = -« + ^' and, integrating again, y = c^ - ax -\- ^,log[^ -f v^(i + ^')]. § VIIL] EQUATIONS IN WHICH X DOES NOT OCCUR. 83 87. In general, an equation of the ;2th order which does not contain y, and in which the lowest derivative is of the rth order, is equivalent to an equation of the {71 — r)th order for the determination of this derivative. For example, is equivalent to Integrating, we have d^q dx^ ^ dx^ and, integrating twice more. Equations in which x does not occur. 88. An equation of the second order in which x does not occur may be reduced to an equation of the first order between y and p by putting d'^y _ d^ ^ dp dy _ dp dx^ dx dy dx dy' For example, the equation dx .djy^/dy\^ ^ ^x^ \dx) ^^ thus becomes dy fP^ ^ P\ (2) or dp _ dy ^ p^~y' 84 EQUATIONS OF THE SECOND ORDER. [Art. 88. whence / y or dx — -^ -\- Cidv'y y and, integrating again, X = \ogy -{- c,y + c^ (3) In equation (2), we rejected the solution / = o, which gives y z=: C \ but it is to be noticed that the equation is still satisfied by / == o after the rejection of the factor /; accord- ingly, y =. C IS 3. particular system of integrals included in the complete integral (3), as will be seen by writing the latter in the form y == A ^ B{x - logj), and making B =■ o. 89. If the equation contains higher derivatives, they may, in like manner, be expressed in terms of derivatives of / with respect to y. Thus, dx^ dxdx^ ^ dyY dyj df ^\dy) In like manner, the expression for the fourth derivative may be found by applying the operation / -7 to this last result, and so ay on. The Method of Variation of Parameters. 90. When the solution of an equation in which the second member is zero is known in the form y =■ f{x)^ the more general equation in which the second member is a function of X may sometimes be solved by assuming the value of y in § VIII.] THE METHOD OF VARIATION OF PARAMETERS. 85 the same form as that which satisfies the simpler equation, except that the constants or parameters in that solution are now assumed to be variables. By substituting for y in the given equation its assumed value, we obtain an equation which must be satisfied by these new variables. When the given equation is of the first order, there is but one new variable, and the method amounts merely to a transformation of the dependent variable ; but when the equation is of the f/th order, the assumption involves ;/ new variables, and we are at liberty to impose ;/ — i other conditions upon them beside the con- dition that the given equation shall be satisfied. The condi- tions which produce the simplest result are that the derivatives of J, of all orders lower than the ;/th, shall have the same values when the parameters are variable as when they are constant. 91. For example, given the equation we assume y = C^Q.o'?>ax + CjSin^^, (2) which, if C^ and C^ are constant, satisfies the equation when X = o. Now, if C, and {Tj are variable, we may assume this value of y to satisfy equatioii (i), and, at the same time, impose a second condition upon the two new variables. Differentiating, we have -^ = —aCi^max + aC2 cos ax H cos ax -| sm ax, dx dx dx in which the first two terms form the value of -^ when C^ and dx C2 are constant. We now assume, as the second condition mentioned above, dCi , dC2 • / X — -0.0% ax H -^Ys\ax = 0, (3) dx dx 86 EQUATIONS OF THE SECOND ORDER. [Art. 9 1. which makes JL — —aCy'SiViax + aC^ cos ax, dx Differentiating again, we have -^ = — a} C ^0.0% ax — a^CiSinax — a — •'sm^.r + a — -cos ax. dx^ dx dx Substituting in equation (i), we obtain — a — ^sin^^^c -f a — ^cos^jc — X . , . . (4) dx dx as the condition that y, in equation (2), shall satisfy the given equation. Equations (3) and (4) give, by elimination, — a — -' = X sin ax, a — ? = X cos ax ; dx dx whence Ci = — Xsin^;c^:r 4- Cj, Cj = -\X cos axdx + ^2; and, substituting in equation (2), y — — cos ax\X sm axdx -\ — sm ax\X cos axdx -H Ci cos ax 4- ^2 sin ax, as otherwise found in Art. 82. The method of variation of parameters is of historic interest as one of the earliest general methods employed. It may occasionally be applied also when the term neglected in finding the form to be assumed for the value of y is not a mere function of x ; but, for the most part, examples which can be solved by it can be more readily solved by the methods given in the succeeding chapters. § VIII.] EXAMPLES. 8y Examples VIII. Solve the following differential equations : — I. -j^ = xe"^, y =i {x — 2)^-^ -\-c^x H-^Tj. ax'* 2. —^ = sin3^, 7 = Jcos'a; — -^cos'^ + c^x^ + c^x + Cy dx^ 4. Find a first integral of yf = /(^-^), (^)' = 2 \f{x)dx. 5. — ^ =~^J£: + ^_>' [3 > o], ~ax ■\- by — Ae"^^ + Be-"^^, d^y flt^i: — ^j' = ^sin^y/3 + BcQ^xsjb, 6. -^ = «^ — ^y f/^ > ol, dx^ ^ ^ -'' dx^ >J{2ey + c^) ^ c ' e-y= V" , or c — ^ 2^:'' = c^%tz'-{\cx + C), according as the first constant of integration is c^, o, or — ^. ^ ^^v I dy\ , »• ^=(i; +'' .,.^= cos (.+..), BS EQUATIONS OF THE SECOND ORDER. [Art. 9 1. ax* X ax dx* ax a — y dx* ax J 13. -^ + MY +1 = 0, ;; = logsin(;c - a) + ^. 14. Show that y— ;i? -^ is an exact differential. ^ -^ dt^ dt* • dx* x^ X 16. (i - x")^ - a:^ =2, y = (sin-^;«:)^ + ^.sin-^;*; + c^. 17. (i - x^)-f- + ^-^ = ^^, ax^ ax y z= ax •\- ^j[sin-^.:v + ^^{^ — •^■^)] + ^2' - <'-^->£+'+(iy=°' § VIII.] EXAMPLES. 89 22. ;;(! - log J)^ + (i 4- log>')(;^j = o» ^3-^£-(i;=>^'o^-' log^ = I + \ log J = c^e^ -\- c^e-^. 2d( 2§. -— - + « = v/(i + /^^sin^^) , ,^ . u = J^-i^ — -^- ^ + ^cos(^ — a). I + ^' 27. Determine the curve in which the normal is equal to the radius of curvature, but in the opposite direction. The catenary 7 = '2 + . . . + Cnyn .... (3) will satisfy the equation ; and, since this expression contains n arbitrary constants, it will be the complete integral of equation (2). Thus the complete integral is known when ;/ particular integrals are known, provided they are distinct ; that is to say, such that no one can be expressed as a sum of multiples of the others. 94. Now let Y denote a particular integral of the more general equation (i), and let u denote the second member of equation (3), that is to say, the complete integral of equation (2). If we substitute y = Y + u (4) in the first member of equation (i), the result will be the sum of the results of putting y =. Y, and y =^ u respectively. The first of these results will be X because F satisfies equation (i), the second result will be zero because u satisfies equation (2) ; hence the entire result will be X, and equation (4) is an integral of equation (i). Moreover, it is the complete integral because // contains n arbitrary constants. Thus the complete integral § IX.] PROPERTIES OF THE LINEAR EQUATION. 93 of equation (i) is known when any one particular integral is known, together with the complete integral of equation (2). In equation (4), Y is called the particular integral, and it is called the complementary function. The particular integral contains no arbitrary constants, and any two particular integrals may differ by any multiples of one or more terms belonging to the complementary function. Linear Equations with Constant Coefficients and Second Member Zero. 95. In the equation dx*" dx"^-^ dx in which the coefficients A^, A, . , . A„ are constants, let us substitute y =■ e^^ where m \^ 2i constant to be determined. ' d d^ Since —e'"'' = me"""", — ^'«^ = m^e*""", etc. ; the result, after ax dx^ rejecting the factor ^'^^j is A^m*' -\- A^m^--^ -\- . , . -\- An-rm -\- An =^ o, . . (2) an equation of the n\h. degree to determine m. Hence, if m j, satisfies equation (2), e*^^ is an integral of equation (i) ; and, if . m^^m^... mn are n distinct roots of equation (2), y = C.e^x^ + C^e^^ + . . . + CnC'^nX ... (3) is, by Art. 93, the complete integral of equation (i). For example, let the given equation be d^y dy — — — -^ — 2V = o : dx' dx ^ ' 94 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 95. the equation to determine m is m^ — m — 2 = o, whose roots are — i and 2 ; therefore the complete integral is 96. Denoting the symbol — by D, equation (i) of Art. 95 may be written {AJ)^ + A,l>-^ + . . . + An-.D -f- An)y = o, or, symbolically, AJ^)y = o, . (I) in which / denotes a rational integral function. With this notation, equation (2) of the preceding article becomes f{m) = o ; and, denoting its roots, as before, by ;«„ 7n^ . , . ntn, equation (i) may, in accordance with the principles of commutative and distributive operations (Diff. Calc, Art. 406 et seq), be written in the form {D - m,) {D - 7n;) . . . {D - m„)y = o. . . . (2) This form of the equation shows that it is satisfied by each of the values of j/ which separately satisfy the equations (Z> — m,)y = 0, (Z) — m2)y = 0, . . . (Z> — m„)y = o ; that is to say, by each of the terms of the complete integral. § IX.] CASE OF EQUAL ROOTS. 95 Thus the example given in the preceding article may be written (i?+ i)(Z>- 2)7 = 0, and the separate terms of the complete integral are the integrals of {D + \)y — o and (Z> — 2)y = o, which are d^"* and Cie""^ respectively. Case of Equal Roots* 97. When two or more roots of the equation f{m) = o are equal, the general solution, equation (3), Art. 95, fails to represent the complete integral ; for, if m^ = nti, the corre- sponding terms reduce to in which C^ + C^ is equivalent to a single arbitrary constant. It is necessary then to obtain another particular integral ; namely, a particular integral of {D - m,yy = 0, (i) in addition to that which also satisfies (D — m^y = o. This integral is obviously the solution of {D — m^y = Ae""^"", (2) for, if we apply the operation D — m^ to 'both members of this equation, we obtain equation (i). Equation (2) is a linear equation of the first order, and its complete integral is = \Adx = ^-m^xy — 14^^ ^ Ax -^ By 96 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 97. or y = e^^'i^Ax + B) (3) Hence the terms of the integral of /(B)j/ = o corresponding to a double root of /(m) = o are found by replacing the constant of integration by Ax + B. For example, given the equation dx^ dx^ dx or D{D - ^Yy = o, the roots of fipi) = o are o, i, i, and the complete integral is y z=C -\- e^{Ax + B). 98. If there be three roots equal to m^^ we have, in like manner, to solve {D - m,yy = o (i) But the integral of this is the same as that of {D — pt^)y = e'^^^iAx + B)) (2) for, by the preceding article, if the operation {D — m,Y be applied to each member of this equation, the result will be (D — mfy = o. The integral of equation (2) is e-m,xy = U^jc + B)dx = iAs^ -\- Bx -^C; t>r, writing A in place of ^A, y = e^i^iAx' + Bx -hC) (3) Hence the terms corresponding to a triple root of /(m) = o are found by replacing the constant of integration by the § IX.] CASE OF' IMAGINARY ROOTS. 97 expression Ax^ + Bx + C. In like manner, we may show that the terms corresponding to an r-fold root m, are e'^i^iAx''-^ + Bx"--^ + . . . 4- Z). In particular, if the r-fold root is zero, we have for the integral of y = Ax"---" 4- Bx"--^ + . . . + Z, as immediately verified by successive integration. Case of Imaginary Roots. 99. When the equation f{m) = o has a pair of imaginary roots, the corresponding terms in the complete integral, as given by the general expression, take an imaginary form ; but, assuming the corresponding constants of integration to be also imaginary, the pair of terms is readily reduced to a real form. Thus, if w, =r a + t(3 and ntz = a — //?, the terms in question are C^^ia + zl8 )x _|. C^^(a - t? )x ^ gax(^ C,e^^^ + C^*? " ^^^) . . . ( I ) Separating the real and imaginary parts of e^^"^ and e-*^^j the expression becomes e^'^liC, + C2)cos/3x + t{C, - C^) sin fix}; or, putting d -{- C2 =■ A and i(Ci — C2) = B, . r^{Acosfix -t BsmjSx), (2) where, in order that A and B may be real, d and C2 in (i) must be assumed imaginary. 98 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 99. As an example, let the given equation be (/?» + /^ + 1)7 = o; the roots are —J ± \i^l ; here a = — J, ^ = \^i ; hence the complete integral is y = e-UAcos^x + ^sin^A 100. If the equation /(m) = o has a pair of imaginary r-fold roots, we must, by Art. 98, replace each of the arbitrary constants in expression (i) by a polynomial of the (r — i)th degree ; whence it readily follows that we must, in like manner, replace the constants in expression (2) by similar polynomials. Thus the equation or {D^ + ^Yy = o, in which ±/are double roots, has for its integral y = (A I + Bix) cos X + (A2 H- B2X) sin x. The Linear Equation with Constant Coefficients and Second Member a Function of x. Id. In accordance with the symbolic notation, the value of y which satisfies the equation AD)y = AT ........ (I) is denoted by ^ = A^/- ^^> § IX.] THE INVERSE OPERATIVE SYMBOL. 99 Substituting this expression in equation (i), we have which may be regarded as defining the inverse symbol (2), so that it denotes any function of X which, when operated upon by the direct symbol f{P), produces the given function X. Then, by Art. 94, the complete integral of equation (i) is the sum. of any legitimate value of the inverse symbol and the complementary function or complete integral of AD)y = o. This last function, which is found by the methods explained in the preceding articles, we may call the complementary function for f{D) ; and we see that two legitimate values of the symbol X may differ by an arbitrary multiple of any term in the complementary function ior f(D) ; just as two values of \Xdx or —X may differ by an arbitrary constant, which is the com- plementary function for D. 102. With this understanding of the indefinite character of the inverse symbols, it is evident that an equation involving such symbols is admissible, provided only it is reducible to an identity by performing the necessary direct operations upon each member. It follows that the inverse symbols may be transformed exactly as if they represented algebraic quantities ; for, owing to the commutative and distributive character of the direct operations, the process of verifying the equation is precisely the same whether it be regarded as symbolic or algebraic. For example, to verify the symbolic identity jy — a^ 2a\D -a D ^ a ) 100 LINEAR EQUATIONS: CONSTANT COEFFICIENTS.[Art. 102. we perform the operation D^ — a' on both members ; thus X = ±Ud + a)(D - a)-l—X- {D - a){D ^ a^—^x'^ 2a[_ D — a iJ -\- a J = ±[{D + a)X - (D - a)x'\ = —2aX = X, 2a[_ J 2a the process being equivalent to that of verifying the equation _i_ = -L/_? L_\ D' - a" 2a\D -a D + aj considered as an algebraic identity. 103. The symbol X denotes the value of y in the equation of the first order hence, solving, we have I -^ - ay =z X; dx X = e^Ae-^^Xdx (i) D — a By repeated application of this formula, we have ! X = — ? — e^Ae-<^^Xdx = e^A\e-^^Xdxdx) (2) {D - ay D - a ] JJ ' ^ ^ and, in general, Tn^.'' = ^'\\--Y'''"^'- • • • (3) the last expression involving an integral of the rth order. § IX.] GENERAL EXPRESSION FOR THE INTEGRAL. lOI General Expression for the Integral. 104. We may, by means of equation (i) of the preceding article, write an expression for the complete integral of f{P)y = X involving a multiple integral of the 7/th order. For, using the notation of preceding articles, we may put f{D) = (Z> - m;){D ^m,),..{D- m„); whence I Y _ I I I Y /(jD) ^ D - m, D - m^" ' D - Mn = ^»«i-^U(w2-/«i)jr . . . , ei^z~^dx\ , , \e-mnXdx^'j. . . (i) but the expression given below is preferable, involving, as it does, multiple integrals only when the equation f(P) = o has multiple roots. 105. Let — — - be resolved into partial fractions ; supposing m^y m^ . . . m„ to be all different, the result will be of the form /{£>) D - m, D - m^ D - ntn in which N^, N^ . . . N^ are determinate constants ; hence, by equation (i). Art. 103, j^X = N,e^Ae-^r^Xdx + . . . + Nne"'nAe-*'^n^Xdx, (2) which is the general expression * for the complete integral * First published by Lobatto, "Theorie des Caracteristiques," Amsterdam, 1837 ; independently discovered by Boole, Cambridge Math, jfournaly ist series. vol. ii. p. 114. 102 LINEAR EQUATIONS: CONSTANT COEFFICrENTS.{Ax\.. 105. when the roots of /(/>) = o are all different ; each term, it will be noticed, containing one term of the complementary function. When two of the roots of f{D) = o are equal, say ;;/, = w^, the corresponding partial fractions in equation (i) must be assumed in the form £> - m, {D - 7n,y' and then by equations (i) and (2), Art. 103, the corresponding terms in equation (2) will be \e*"^^ e - ""i^Xdx -\- N^e""^^ , iV;^*"!-^ e - "'I'^Xdx 4- N^e'"^^ \\e- '^'^''Xdxdx. In like manner, a multiple root of the r\.\\ order gives rise to multiple integrals of the rth and lower orders. 106. When f{D) = o has a pair of imaginary roots, a ± 2/?, we may first determine, for the corresponding quadratic factor, a partial fraction of the form {D - ay + fi- The corresponding part of the integral will be found by applying the operation N^D H- N^ to the value of {D - ay + ^' X. Decomposing the symbolic operator further, this expression becomes 2i^\D - a - /yS n - a + ip) ' that is. 2/^3 J 2/^ J § IX.] EXAMPLES. 103 This last expression is the sum of two terms of which the second is the same as the first with the sign of i changed ; and, the first term being a complex quantity of the form P -f iQ where P and Q are real, the sum is 2/*, or twice the real part of the first term. Hence (Z> - a)- 4-/32 = the real part of — (cos /Sjx: + /sin/?jc) U-«^(cosjS^ — i€m.px)Xdx, or I {D - ay -{- 13^ When a = o, this result reduces to that otherwise found in Arts. 91 and 82. Examples IX. Solve the following differential equations : — I. ^ - 5 ^ + 6)^ = o. y = c,e^^ + c^e-i^, dx^ ax 2. 6—^ =z -2 + y^ y — c^e^x + c^e-\' 3. ^'^(^ ■\-y\=- (a' + ^)^, > - ^/' + ^a/'. 104 I^r^^^EAR EQUATIONS: CONSTANT COEFFICIENTS\Kx\.. I06. 4. --^ — 2^ + SD' = o, _y = e'{^Aco%2x H- -5 sin 2^). y = ^,^J^ + c^e-^ H- ^sin|(^ -f- a). 6. ^ + ^-6^ = 0, ;- = ....^ + .,.-3. + .,. ; -f- e^{c^ + ^4^)cos;t. ^ dx" ^ , . . x'^xnax , co%ax\oQ,Q.o's>ax y =i CiCosax + CzSmax H H -2 . a a^ § IX.] EXAMPLES. 105 14. -^ Ar y ^ sec^^, ^ = ^ cos (^ + a) + sin:vlog^^t_^HLf _ i. cos^ (Z> - a)^ + ^- — (a cos /?^ — * /? sin yS^) L - «-»^ sin fSxXdx 1 7. Show that ^ J^ — I %max\cos axXdx — cosa^ : 2^1 J J COS axXdx — cos ax sin axXdx --i-r — ^ cos ^x COS axXdx^ + sin «^ sin axXdx^ . y = ^i^-2-=p + ^^(^jcosjc + rjsin:^) + LJl\e'2xXdx 10 J — (3sin;i: — cos:v)L?--*cos;i;X/;k: — (3 cos X 4- sin x)\e-^ sin xXdx \. ^ 15. ^ + y = tanx, v = ^ cos (^ + a) - cos ^ log ^ "^ ^^"'^ . dx^ ZQSX (■ 16. Show that ^ ^ I06 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. lO/. Symbolic Methods of Integration. 107. The foregoing general solution of linear equations with constant coefficients, Art. 105, is theoretically complete ; for the solution of a differential equation consists in finding a relation between x and y involving only the integral sign. But, in the case of certain forms of the function X of frequent occurrence, while the evaluation of the integrals arising in the general solution would be tedious, the final result may be very expe- ditiously obtained by the methods now to be explained. In the first place, suppose the second member X to be of the form e^^ \ in other words, let it be required to solve the equation f{D)y = e-- (i) Since, as in Art. 95, D^'e^^ — aTe^'', and f{D) is a sum of terms of the form Aiy, f{D)e-- = f{a)e-- ; (2) whence Here f{a) is a constant ; and therefore, except when f{a) = o, we may divide by it and write ^" = -^^^ (3) which is the value of y in equation (i). Thus we may, when the operand is of the form Ae^^, put D ^=^ a 'm the operating symbol except when the result would introduce an infinite coefficient § X.] SYMBOLIC METHODS OF INTEGRATION. 10/ io8. In the exceptional case, equation (2), of course, still holds ; but it reduces to f{D)e^^ = o, and thus only expresses that e^^ is a term of the complementary function. In this case, we may still put a for D in all the factors of f{D) except D — a. Thus, putting we have /{D) D - a ) {a) Again, if f{D) = {D ^ ay(D), so that ^ is a double root of f{D) = o, we shall have ■AD) {D - a)^ {D)'^ cf>{a)' JJ^'^ 2{a) 109. As an illustration, let it be required to solve the equation g _;; = (..+ I). (I) The complementary function is The particular integral is In the first and third terms, we may put D =. 2 and o I08 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. IO9. respectively, thus obtaining |^'-^ — i ; but D = i makes the second term infinite ; hence we write The complete integral of equation (i) is therefore y = e-^^(Acos^x -f Bsin^x\ + e^(C + |^) + ^e^"" - i. ' no. The value of the particular integral in the case of failure of equation (3), Art. 107, may also be derived directly from that equation by the principle of continuity. It must be remembered that properly the equation should be understood to contain the complementary function in the second member. Hence, a being a root of /{D) = o, and at first assuming the operand to be e^"^ + ^^"^^ we may write Developing ^^, the second member becomes in which the first term is part of the complementary function. We may therefore write, for the particular integral, i^{a Jr h) \ 2 I because, a being a root of f{z) = o, f{z) =z {z — a)(f>{z), and f{a 4- h) = /i{a -h h). § X.] SYMBOLIC METHODS OF INTEGRATION. IO9 Now, making h := om. this result, we obtain L_ ^ax — \ y.gax AD) cl>{a) ' as before. This is an instance of a general principle of which we shall hereafter meet other applications ; namely, that, when the par- ticular integral, as given by a general formula, becomes infinite, it can be developed into an infinite term which merges into the complementary function, and a finite part which furnishes a new particular integral. Again, when <^ is a double root, and X = ^% the infinite expression can be developed into two infinite terms which merge into the complementary function, together with a finite term which gives the new particular integral. For example, since h is ultimately to be put equal to zero, we may write gax -— ^(a + A);r {D ~ ay{D) <}>{a + /i)k' gax l^\ 2 / (n) 2cl>{a)' Case in which X contains a Term of the Form sin ax or cos ax, III. We have, by differentiation, D sin ax = a cos ax, D^ sin ax =^ —a^ sin ax, lf*'€\Viax— (^ — a^Y €v[i ax \ whence f(^iy)^m.ax — /{ — a'') sin ax. 1 10 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. III. and, in like manner, we obtain f{I>) cos ax — /(—«») cos ax. It follows, as in the similar case of Art. 107, that and _L_cos<.^ = ^^cos«^, W except when/(— ^*) = o. It is obvious that we may include both these results in the slightly more general formula j^—'SiVi^ax + a) ^ ^ €\xi{ax + a). For example, to solve ^ - >' = sin(x + a), we have, for the particular integral. ^ ^_ ^ sin {x -h a) = -J sin (a: + a), Adding the complementary function, we have the complete integral y = c^e"" + c^.e-'' — \€\n{x + a). 112. In order to employ equations (i) and (2) when the inverse symbol is not a function of D^, we reduce it to a fractional form in which the denominator is a function of § X.] SECOND MEMBER OF THE FORM ?AXiax OR Q,Q<$,ax. Ill Z>^ This is readily done ; for we may put f{D) in the form f,{D^) -f Df^{D^)y and the product of this hy f,{D^) - Df^{D^) will be a function of /?^ Moreover, since we have ultimately to put D^ = —a'y we may at once put —a^ in place of D^ in the expression for /(/?), which thus becomes For example, given the equation {J> + D - i^y ^ sin 2^; the particular integral is I I . D + 6 . sm 2x = sm 2x = ■ sm 2x I> -^ D - 2 D-d ^'-36 = _ -^ + ^ sin 2X = - cos 2-^ + 3 sin 2x ^ 40 20 Adding the complementary function y = C,^^ + C,.— - cos2^ + 3sin2^^ 20 113. The case of failure of the formulae (i) and (2) of Art. Ill takes place when the operand is a term of the complementary function. Thus, if the given equation is -^ -\- a^y z=. 0,0^ ax. dx" ^ the complementary function is A cos ax -\- B sin ax. Accord- ingly, in the particular integral -—^ — ^ cos ax, the substitution D^ =. — a^ gives an infinite coefficient. The most convenient 112 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 1 1 3. method of evaluating in this case is that illustrated in Art. no. Thus, putting a -\- h for a in the operand, and developing cos (ax + hx) by Taylor's theorem, ^ cos (a + K)x I / . h}x^ \ = f cos ax — sin ax ,hx — cos ax . h • • • |. -{a-\- hy + a\ 2 I Omitting the first term which belongs to the complementary function, we may write, for the particular integral, I / . hoc^ \ cos (a 4- h)x = -( x^max •\- — cos «x — . . . ) ; 2a -{- h\ 2 / D" + a and, making /r = o, we obtain I X sin ax cos ax = , D^ + a^ 2a and the complete integral of equation (i) is y =z A cosax + B cosax + £_!1?L^. 2a Case in which X contains Terms of the Form X^. 114. If an inverse symbol be developed into a series pro- ceeding by ascending powers of D, the result of operating upon a function of x with the transformed symbol is, in general, an infinite series of functions ; but, when the operand is of the form ;r'«, where m is 3. positive integer, the derivatives above the mth vanish, and the result is finite. For example, to solve -^ + 2y = x^, ax § X.]- SECOND MEMBER OF THE FORM 'X"^. II3 the particular integral is Z> + 2 2 1 + ID = i(i -^D + \D^ _ iz)3 + . . .);,3 1 — ^ (^3 _ 3^2 4_ |^_|). and the complete integral is J = C^-2^ + \x^ - \x^ -I- 3^ _ 3, This result is readily verified by performing upon it the opera- tion D -\- 2. 115. When the denominator of the inverse symbol is divis- ible by a power of D, the development will commence with a negative power of Z>, but no greater number of terms will be required than would be were the factor D not present. For example, if the given equation is (Z)4 + Z>3 + D^)y = ^3 + ^x"^ the particular integral is Since the operand contains no power of x higher than x"^, it is unnecessary to retain powers of D higher than D^ in the development of the expression in brackets. Hence we write •>' = i-.(^ - ^ + ^')(^' "^ ^^^ = (i^ ~ 5 ■*" ^)(^' ^ 3^'^ X^ . X^ X* , , , , , = 1 X^ -^ 2X^ + 6x, 20 4 4 1 14 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 1 1 5. in which the last term should be rejected as included in the complementary function. Thus the complete integral is y = x^ + xx^ -\- CiX -^ C2 + e-^^ic, cos^^ + ^r.sin^^j. "^20 \ * 2 2 / It will be noticed that, had we retained any higher powers of D in the final development, they would have produced only terms included in the complementary function. Symbolic Formulce of Reduction, 116. The formulae of reduction explained in this and the following articles apply to cases in which X contains a factor of a special form. In the first place, let X be of the form e^* V, V being any function of x. By differentiation, ±e<'xv = ^^^ 4. ae^^F, dx dx or De^^V = e^^{D -\- a)V. (i) By repeated application of this formula, we have J>e^^V = De^^{D + d)V = ^-^(Z) + ayV\ and, in general, jye^xy = ^^(2) + ayV, Hence, when <^(Z>) is a direct symbol involving integral powers of Dy we have (l){D)e^^F = e^^cjyiD + a)K (2) § X.] SYMBOLIC FORMULA OF REDUCTION. II5 To show that this formula is applicable also to inverse symbols, put whence V = ^ Y • and equation (2) becomes in which Fi denotes any function of x^ since V was unrestricted. Now, applying the operation to both members, we have which is of the same form as equation (2). As an example of the application of this formula, let the given equation be -^ — V = xe^^. The particular integral is y = ? e'^^x — e"*^ ? x I e^"^ I X = r (I - iz) + . . .)^ = ^ - ^^ 3 39 1 16 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 1 1/. 117. The formula of reduction of the preceding article may often be used with advantage in the evaluation of an ordinary integral. For example, to find Ic"*"^ sin fixc/x, we have, by the formula, -^ e*»^ sin ?tx = e^^ — -^ — sin nx : '■'■ D D + m ' hence ^'«-* sin nxt/x = ^'"•^ ~ ^ sin nx J D^ - ni" {m — D) sin nx = {m sin nx — n cos nx). m^ -\- n^ m^ + n^ It may be noticed that equation (i), Art. 103, is a case of the present formula of reduction, for — I — X = — l—e^^e-^^X] D - a D - a hence, applying the formula, we obtain — -^ X — e^^—e-^^X — e^Ae-^^Xdx\ D - a D J ' in which we pass from the solution of a differential equation to a simple integration. In the above example, on the other hand, we employed the same formula to reverse the process, the direct solution of the differential equation being, in that case, the simpler process. Compare Int. Calc, Art. 63. 118. Secondly, let X be of the form xV. By successive differentiation, we have DxV = xDV -f F, D^xV = xD^V + 2Z>F, D^x V = xD^ V -f- 3Z>2 V; and, generally, lyxV = x]>v -h riy-^v. . ... . (i) § X.] SYMBOLIC FORMULA OF REDUCTION. 11/ Now let <^{P) denote a rational integral function of D, that is, the sum of terms of the form a^D'' ; and let us transform each term of ^{D)xV by means of equation (i). We thus have two sets of terms whose sums are x^ayD'^V and ^a^rD^'-'^V respectively. The first sum is obviously ;ir0(Z)) F; and, since ayrD""-^ is the derivative of a^D^ considered as a function of D, the second sum constitutes the function <^'{P) V, Hence (f>(jD)xF = x(l>{D)V -{- (l)\D)V, .... (2) where <^' is the derivative of the function <^. To show that this formula is true also for inverse symbols, put whence F = — ^ V, : and equation (2) becomes *(^)^^^' = ^^' + *'(^>^^" or xF = 6(D)x xF, = ct){D)x-^~F, - ^i^K, in which Vj denotes any function of x. Hence, applying the operation — — - to both members, we have the general formula -^-xF= x~^F-^^^F,. ... (3) which is of the same form as equation (2), because — — is the derivative of the function — . Il8 LINEAR EQUATIONS: CONSTANT COEFFIC I ENTS\hx\.. 1 1 9. 119. As an example, take the linear equation dy ax By the formula, the particular integral is ^sin^=jc— sinjc — ^ — — sinj: Z> - I D - \ {D - 1)' = X — — %mx — ^-— 9in^: hence y = — ^.r(cos^ -f- smx) — ^cos^ + Ce^. This example is a good illustration of the advantage of the symbolic method, for the general solution would give the integral in the very inconvenient form ,y = ^^j, e-^xsmxdx + Ce^ , and, in fact, the best way to evaluate the indefinite integral in this expression is by the symbolic method, as in Art. 117. 120. Finally, let X be of the form x^ V. Putting ;ir F in place of V in formula (2), Art. 118, {D)x^V = xti>{D)xV + <\i'{D)xV) and, reducing by the same formula the expressions (f>{D)x V and \D)xVy this becomes (Z>)x^V = x^(j>{D) V + 2X(j>\D) V + ^"{D) V. . (4) Again, putting xV 'for V in this formula, and reducing as before, we have {D)x^V= x^{D)V + 2,x^\D)V+ zx"{D)V+ <\>"\D)V; and by the same process we obtain similar formulae for r^V, x^V, etc., the numerical coefficients introduced being obviously those of the binomial theorem. § X.] SYMBOLIC FORMULA OF REDUCTION. 1 19 As an illustration, let the given equation be By formula (4), the particular integral is x' sm 2x D^ -h I 2Z> . , 6Z)2 — 2 . sm 2;tr + 2x sin 2^ -\ — sin 2x D^ + 1 (Z)2 + i)" (i:)" + 1)3 x^ 8jc 2 6 • — sin 2;t: cos 2x -\ sin 2x, 3 9 27 and the complete integral is )x^ — 26 . 8jc , . QJC^ — 20 . iSX y z= Ci cos ::!£: + ^2 sin jc — 2 sin 2:x: cos 2X. 27 9 Employment of the Exponential Forms of sin ax and cos ax. 121. It is often useful to substitute for a factor of the form ^vaax or cos^;r its exponential value, and then to reduce the result by means of formula (2) of Art. 116. For example, in solving the equation — ^ -f y = x^ sin X, dx^ we have, for the particular integral, V = jc^sin^ = ie^^ — ^-'*-*^): but it is rather more convenient to write, what is easily seen to be the same thing, since e*'^ = cos;ir + z sin;ir, y = the coefficient of / in a:V-*. 20 LINEAR EQUATIONS: CONSTANT COEFFICIENTS. [Art. 121. Now I x^e'^ — e'^-— : x^ = e'^- 21 D\ 21 4/^ / = (cosAT + i^mx^i-— + ^ + — ^; \ 6 4 4/ whence, taking the coefficient of /, and adding the complement- ary function, Examples X. Solve the following differential equations : I. i-2 — y = a:^2^ 4- e^, dx^ y = c,e^ -\- c^e-^ + — (3-^ - 4) + ~- * This method has an obvious advantage over that of Art. 120 when a high power of X occurs. Moreover, when, as in the present example, the trigonometrical factor is a term of the complementary function, it should always be employed. For it is to be noticed that, in formula (3), Art. 118, while two legitimate values of the symbol in the first member can differ only by multiples of terms in the comple- mentary function of ^{D), two values of the second member may differ by the product of one of these terms by x. Hence a result obtained by the formula might be erroneous with respect to the coefficient of such a term. In the example of Art. 119, the uncertainty would exist only with respect to a term of the form xex, but it is easy to see that no such term can occur in the solution. In the example of Art. 120, a similar uncertainty exists with respect to terms of the form jr^sin jt, x^Q.o%x, x%\xix, and x cosjt, none of which occur in the solution. In the present example, if solved by the same method, the uncertainty would exist with respect to terms of the same form ; and, as such terms do occur in the solution, an error might arise. See Messetiger of Mathematics, vol. xvi. p. 86. § X.] EXAMPLES. 121 dv 2. -^ — 2y =^ x^ + e^ -\- cos 2^, ^^ y = - = ^^{x^ + a:0 + C. F. ax* ax 144 ^+2^ ^/a:^ ^;»^ y = h ^ cos ^ M- C. F. 12 48 15. -^ — 2-/ + 4;; = ; = ^2 ^ x-\ y = e-^{c^ + CiX + Cjpc^) -{- x^ — 6x + 12 -j- e- dxK 17. (D -\- b)*'y = cos^^, 4- {a^ + ^2)2cos(«jc — «cot~'-]. 18. Expand the integral \oc"e^dx by the symbolic method. y.x»e^ = e^lx*' — nx»-^ + n{n — i)x»-'' — ...]+ c. 19. Prove the following extension of Leibnitz' theorem : — {D)uv = u . (l){D)v + J^u . )z; + — • <^''(^)z/ + . . . , 2 ! and show that it includes the extended form of integration by parts, Int. Calc, Art. 74. § X.] EXAMPLES. 123 20. In the equation connecting the perpendicular upon a tangent with the radius of curvature, (Diff. Calc, Art. 349), / and <^ may be regarded as polar coordinates of the foot of the perpendicular. Hence show that, if the radius of curvature be given in the form p = /(), the equation of the pedal is r=^cos(0-|-a)+^^^-l— /(^), and interpret the complementary function (W. M. Hicks, Messenger of Mathematics, vol. vi. p. 95). 21. The radius of curvature of the cycloid being p = 4«cos(^, find the equation of the pedal at the vertex. r = 2aQ sin Q, 124 LINEAR EQUATIONS: VARIABLE COEFFICIENTS\hx\., 122. CHAPTER VI. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS. XL The Homogeneous Linear Equation, 122. The linear equation dx*^ dx» - ^ in which the coefficient of each derivative is the product of a constant and a power of x whose exponent is the index of the derivative, is called the homogeneous linear equation. The operation expressed by each term of the first member is such that, when performed upon x"^, the result is a multiple of x"^ ; hence, if we put y ■=^ x*^ va the first member, the whole result will be the product ♦of ;r"' and a constant factor involving m. Supposing then, in the first place, that the second member is zero, the equation will be satisfied if the value of m be so taken as to make the last-mentioned factor vanish. For example, if, in the equation X^-^ -{■ 2X^ - 2y ^ o, (i) dx^ dx we put y =. X*", the result is \m{m — i) -\- 2m — 2']x^ = o; § XI.] THE HOMOGENEOUS LINEAR EQUATION. 1 25 hence, if m satisfies the equation m^ -^ m — 2 — o, (2) x^ is an integral of the given equation. The roots of equation (2) are i and —2, giving two distinct integrals; hence, by Art. 93, y = CyX + C^X-"^ is the complete integral of equation (i). The Operative Symbol &. 123. The homogeneous linear equation can be reduced to the form having constant coefficients by the transformation x =. e^. For, ii x = e^, we have (Diff. Calc, Art. 417) ti d ^d^ did \ X— = — , x^— = — ( — — I J ; dx de dx" dd\de and, in general, d dxr did \ Id ^ \ so that in the transformation each term of the first member of the given equation gives rise to terms involving derivatives with respect to B with constant coefficients only. Denoting — by D, uu the equation is thus reduced to the form /{D)y = o (i) in which/ is an algebraic function having constant coefficients. 126 LINEAR EQUATIONS: VARIABLE COEFFICIENTS\Kxl. 1 23. Now, if we put ^ for the operative symbol x-— , the trans- (IX forming equations become dx do^ and, in general. ^£: = ^(^- i)(^- 2)...(^-r+ i); and the result of transformation is f{^)y = 0,* (2) in which / denotes the same function as in equation (i), but x is still regarded as the independent variable. As an example of the transformation of an equation to the form (2), equation (i) of Art. 122 becomes [^(^ - i) -\- 2& — 2]y = o, or (&^ + O — 2)y = o. 124. The operator & has the same relation to the function X"' that D has to e"'^ ; for we have ^x'" = mx»*, O^x"" = ;//2^'«, . . . ^''x**^ = m^x^ ] whence f{&)x*^ = /{m)x'" (i) / /V * * The factors x and — of the symbol x — are non-commutative with one dx dx another, and the entire symbol, or iJ, is non-commutative both with x and with D ; but it is commutative with constant factors, and therefore is combined with them in accordance with the ordinary algebraic laws. § XI.] THE OPERATIVE SYMBOL &. 12/ Thus the result of putting j/ = x"" in the homogeneous linear equation /W>' = o (2) is /{m)x^ = o ; whence Am) = o (3) Accordingly, it will be noticed that the process of finding the function of m, as illustrated in Art. 122, is precisely the same as that of finding the function of -d-, as illustrated in Art. 123. If, now, the equation /(m) = o has n distinct roots, m^j m^ . . . w«, the complete integral of f{^)y = o is y = C^^'^i + C:,x'"'^ + . . . + C«^'«« ; .... (4) the result being the same as that of substituting x for e"^ in equation (3), Art. 95. Cases of Equal and Imaginary Roots. 125. The modifications of the form of the integral, when y(^) == o has equal roots, or a pair of imaginary roots, may be derived from the corresponding changes in the case of the equation with constant coefficients. Thus, when f{d) = o has a double root equal to m, we find, by putting x in place of ^^, and consequently log jir in place of x, in the results given in Art. 97, that the corresponding terms of the integral are x'"(A -\- Blogx). In like manner, when a triple root equal to m occurs, the cor- responding terms are x^lA + Blogx + C{logxy^, and so on. 128 LINEAR EQUATIONS: VARIABLE COEFFICIENTS,[Ar\.. 1 25. Again, when f{d) = o has a pair of imaginary roots, a ± y8/, we infer, from Art. 99, that the corresponding terms of the integral may be written The Particular Integral. 126. The homogeneous linear equation, in which the second member is not zero, may be reduced to the form The complementary function, which is the integral of f{d)y = o, is found by the method explained in the preceding articles. The determination of the particular integral, which is sym- bolically expressed by — X, may, by the resolution of — — into partial fractions, be reduced to the evaluation of expression of the form — ^—X, 1 X, etc. & - a ' {& - ay Compare Art. 105. The first of these expressions is the value of y in the equation (^ - a)y = X, or "I - "-^ = ^' a linear equation of the first order, whose integral is x-'^y = Xx-'^-^Xdx) § XL] THE PARTICULAR INTEGRAL. 1 29 hence — ^—X = x'^lx-'^-^Xdx (i) Again, applying the operation to both members of this ^ — a equation, and reducing by means of the same equation, 1 X — — ^ — x^Xx-'^-'-Xdx = xAx-Ax-'^-'^Xdx^ \ (2) {^ - ay ^ - a ] J J ' ^ ' and, in general, ^— — X = x4x-Ax-A . . . \x-''-^Xdx*'. . . (3) i&-a)r J J J J ^^^ 127. Methods of operating with inverse symbols involving & applicable to certain forms of the operand X, and analogous to those given in the preceding section for symbols involving D, might be deduced. The case of most frequent occurrence is that in which X is of the form x^. From equation (i), Art. 124, it follows that, except when /{a) = o, In the exceptional case, « is a root of /(^) = o, and f{%) is of the form (i9- — aY^{^) where <^{a) does not vanish ; hence I I _ _i I {^ ^ ay'^i^) ~ i\>{a) (^ - ay But, by formula (3) of the preceding article, 130 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. 12/. As an example, let the given equation be iif* -^ 4- 2x-^ ^ 2y = x^ + x: dx* dx the complementary function was found in Art. 122 ; for the particular integral, we have I - , I I •^ ^ + ^—2 ^— I^-t-2 ^3 I I _ -^^ I x[dx 1 — — X — — i — . 10 3^—1 10 2>] ^ Hence y = c^x -^ c^x-"* + A^3 + l^log^^c. Symbolic Solutions. 128. The first member of any linear equation may be written symbolically AD, x)y. In the case of the linear equation with constant coefficients, the operator is a function of D only. In the case of the homogeneous equation considered in the preceding articles, the operator is capable of expression as a function of the product xD which we denoted by ^. Examples occasionally occur in which /(Z>, x) admits of expression as a function of some other single symbol which, like ^, involves D only in the first degree. In these cases, the equation is readily solved in the manner illustrated below. Given the equation S _ 2bx^ + b^x^y = o. dx^ dx § XL] SYMBOLIC SOLUTIONS. 13! Since {D — bx){Dy — bxy) = By — bxBy -^ by — bxDy + b'^xy, the equation may be written in the form (Z? — bxYy -ir by ^ o\ or, putting t^iox D — bx, {V + b)y = o, in which the operator is expressed as a function of C Resolving it into symbolic factors, we have and the two terms of the integral satisfy respectively the equations {K — i^b)y — o and (C + isib)y = o. The first of these equations gives {D — bx — i\Jb)y^ = o, or ^ = {bx + islb)dx', and, integrating, logy, = ^bx^ + lybx + ^i, or yi = C,e'^^^^(cosx^b + isinx^b). In like manner, the second equation gives J2 = C2e'^^^^(cosx)Jb — isinx^b). Adding, and changing the constants, as in Art. 99, we have, for the complete integral, y = eh^^ {A cos x\Jb 4- Bsinxsjb). 132 LINEAR EQUATIONS: VARIABLE COEFFICIENTS\hx\.. 128. When the second member of the given equation is a function of Xy the particular integral is found by resolving the inverse symbol into partial fractions, as in Art. 126, the evaluation of each term depending on the solution of a linear equation of the first order. 129. The symbolic operator can sometimes be resolved into factors which are of the first degree with respect to D, but are not expressible in terms of any single operating symbol. In these cases, the factors are non-commutative ; the equation can still be solved, but this circumstance materially alters the mode of solution. For example, the equation g - (^^ + ^)| + (^' -2x)y = X . . . (I) may be written in the form {D-x){D-x^)y^X', ..... (2) for, by differentiation, (£ - ")(l - "'•^) = S - "'I - ^^-^ - "I + "'^- The complementary function satisfies the equation {D-x){D- x^)y=^o', (3) and it is evident that the solution of (Z) - x-)y = o, * which is jK = Ce^'^y satisfies equation (3). But, since we cannot reverse the order of the symbolic factors, equation (3) is not satisfied by the solution of {D — x)y = o. § XL] NON-COMMUTATIVE SYMBOLIC FACTORS. 1 33 130. To solve equation (2), put (Z> - x^y = z;; ....... (4) then the equation becomes {D-~x)v^X, (5) a linear equation of the first order for v. Solving equation (5), we have and, substituting in equation (4), we have, by integration, ;; = e\Ae-\^'^^lAe-hx-'Xdx' + c,e\^^ e-l^^ + h^^dx + c^e\^., (6) 131. The solution of the general linear equation of the first order (^D^- P)y^ X may (see Art. 34) be written in the symbolic form " X = e-l'^Ael'^'^Xdx, jD -{- P which includes the complementary function since the integral sign implies an arbitrary constant. In accordance with the same notation, the value of j, in equation (2), would be written ^ ~ D - x^D -X ' which is at once reduced to the expression (6) by the above formula. It will be noticed that the factors must, in the in- verse symbol, be written in the order inverse to that in which they occur in the direct symbol. 134 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. 131. In obtaining this solution, the non-commutative character of the factors precluded us from a process analogous to the method of partial fractions, Art. 105 ; we have, in fact, only a solution analogous to equation (i) of Art. 104. Examples XI. Solve the following differential equations : — ax^ dx^ dx X 3- 2^" -| + zx-^ - 2>y = X, dor dx y = c,x + c,x-i + i^[^ - ix-^{xhXdx. 4. x^-^^ — 2^ = 0, y = c,x^ + ^2 + ^jlog^. 5. x^—^^ + 4x-^-\-2y = e^, y = c,x-^ + ^^^-^ + x-^e^. 6. ;e^ + sx"^ + 2^ = X, dx^ dx^ dx y = A cos logx -\- B sin log ji; + C -f j^x^. >' = x{A + ^logjc) + Cjc-^ + }^-MogJc. 8. ^^ + 6^'^ + 9^^ +3^1' + ^= 4;., dx* dx^ dx^ dx y — (c^ -\- C2 log x) cos log X + {c^ -^ C4 log ^c) sin log x + x. § XL] EXAMPLES. 135 9. x'^—^ + 2X^-^ 4- 2j/ = \o[x -\--\ dx^ dx^ \ xj y = ^(^coslog;*: + B^mXogx + 5) + x-^{C + 2logA:). 10. x"^—^ + 2x'^—^ — jt^ +_>' = xXogx, dx^ dx^ dx J. = £i + ^r.. + c,\ogx - te£)! + ('°g^)n. ^ L 8 12 J 11. ^ + 4-^^ + 4-^;' = 0, y = ce-^""-^^ + ^^^-^ +W2. 12. Prove that d^x'^Xogx = r^x^logx + nr^-'^x^, 13. Prove that both when /(^) is a direct, and when it is an inverse, symbol. XII. Exact Linear Equations. 132. Using accents to indicate the derivatives of y with respect to x, the linear equation of the «th order is />,/«) + p^yin-^) + . . . 4. P^_^y" + p^_J + p^^ = X (l) To ascertain the condition under which this represents an exact differential equation, and to find its integral when such is the case, we shall employ an extension of the method of Art. 84, which consists in successive subtractions of exact derivatives so chosen as to reduce the order of the remainder at each step 136 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. 1 3 2. until we arrive at a remainder which is, or is not, obviously exact. Since the second member of equation (i) is a function of X only, the equation is exact if, the subtractions being made from the first member, the coefficient of y vanishes in the remainder of the order zero, which contains no derivatives of y. When this condition is fulfilled, the sum of the expressions whose derivatives have been subtracted will be equal to \Xdx + C. The first term of equation (i) shows that the first of these expressions is Poy^*^-^^, of which the derivative is Subtracting this from the first member, the remainder is {F, - Po')y^"-'^ + P^y^"-'^ + . . . + l^y; or, putting Q, = P,- P,\ Q^y{n-i) + p^yin-:^) + . . . + P„y. In like manner, the next expression whose derivative is to be subtracted is Qiy^"~''\ the next remainder being and so on, the values of Q2, Q^y etc., being • Q2 = P2- Q/, Qs = P3- <2/, etc. ... (2) The final remainder is Q„y ; and the condition of exactness is that this shall vanish, that is to say, Q„ = o. If this condition be fulfilled, the integral will be Q^yin-i) + Q^y(n-2) + . . . + Q„_^y + Q^_^y = {xt/x + C (3) § XII.] EXACT LINEAR EQUATIONS. 137 where Q, = P,, Q, = P, - PJ, Q^ z= P, - P/ + PJ\ and in general, Q.^Pr- Pr-^ + Pr'-^ - . . . ± Po^""^ l and the condition of direct integrability written at length is Qn= Pn- P./-X + Pn'-. - . . . ± Po^"^ = C . . (4) 133. For example, to determine whether the equation is exact, we have, by the criterion, equation (4), ^3 = 4 - 14 + 16 - 6 = o; hence the equation is exact ; and, forming the successive values of the coefficients Q by the equations (2), we find ^^' - ^>£ + ^5^ - ^^2 + '^^-^ = -^ + '" which is a first integral of the given equation. Again, on applying the criterion to this result, we obtain 4x — lox -{- 6x = o ; hence it is also exact, and its integral is found, by the same process, to be I (x^ — x)-^ + (2X^ — i);; == i + c,x + ^2, ax x in which a second constant of integration is introduced. This last result is not exact, for 2x^ — i — (2>^^ — i) is not equal to zero ; but it is a linear equation of the first order, 138 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. 1 33. and its solution gives for the complete integral of the given equation, xy>J{x* — i) = scc-^a: + ^V(-^' — i) + c^\o%{x + v^(^ - i)] + c^ 134. The condition of direct integrability, equation (4), Art. 132, contains the rth derivative only of the coefficient of the rX\i derivative of y in equation (i) ; whence it is evident that the product is an exact" derivative when s is a positive integer less thaji r. For example, x^D^y is exact, because the^ourth derivative of x^ is zero ; its integral is x^D^y — sx^£>y + 6xDy — 6. When s is negative, fractional, or an integer equal to or greater than r, a term of the form xWy, in equation (i), gives rise, in equation (4), to a term containing x^-^. From this it is evident that, if, in the given equation, we group together the terms of the specified form in such a manner that s — r has the same value for all the terms in a group, it is necessary, in order that the equation may be exact, that each group should separately constitute an exact derivative. If a single group be multiplied by x*", and equation (4) be then formed, we shall have an equation by which m may be so determined that the group becomes exact ; but, when the given equation consists of only one group, it becomes a homogeneous linear equation when multiplied by x''-', and it is more readily solved by the methods already given for such equations. 135. When an equation containing more than one such group of terms is not exact, it may happen that each group § XIL] INTEGRATING FACTORS OF THE FORM x"". 1 39 %. becomes exact when multiplied by the same power of x. For example, the equation 2x^{x+i)^^+x{^x + z)^-. zy = x. . . (i) ax^ ax contains two groups of terms, in one of which s — r z=: i, and in the other s — r =■ o. Multiplying by x^, and then substi- tuting in equation (4) of Art. 132, we have — 3^^ — "j^m + 2)x^ + ^ — ^{m -f- i)x^ -h 2(m + $)(m -h 2)x^ + ^ + 2{m -\- 2)(;« -f- i)x^ — o, which reduces to {m + 2){2m — i)x^ + '^ -\- {m + 2) (2m — i)x^ = o, . (2) the two terms in this equation respectively arising from the two groups in equation (i). If, now, the value of m can be so taken as to make each coefficient in equation (2) vanish, equation (i) becomes exact when multiplied by x'''. In this instance there are two such values of m ; namely, —2 and ^. Using the first value of m, we have the exact equation a(. + i)^{ + f 7 + 2)1 - % = ^, ax^ \ xjdx x^ x^ whose integral is and, using the second value, we have the exact equation 2{xi + ^t)g + (7^^ + 3^^)^ - 3^b = x^X, whose integral is 2x^(x 4- i)-i^ — 2xb = \x^Xdx — C2. . . . (4) ax J 140 LINEAR EQUATIONS: VARIABLE COEFFICIENTS, [Art. 1 35. fe Having thus two first integrals of equation (i), its complete integral is found, by elimination of y from equations (3) and (4), to be 5(^ + i)>' = c^x H- c^x-^ -f xV^^dx — cxr-MjciAT^. . (5) Symbolical Treatment of Exact Linear Equations. 136. The result of a direct integration is, when regarded symbolically, equivalent to the resolution of the symbolic operator into factors, of which that most remote from the operand y is the simple factor D. For example, the two successive direct integrations effected in Art. 133 show that (^3 _ x)D^ + (8^2 - 3)Z>2 + \^xD -f 4 = Z>2[(jt:3 - x)D + 2^-1]; and, from Art. 135, we infer the two results, 2x\x + i)D^ + x{_']x -t- 3)Z> - 3 = x'D{_2{x 4- i)i) H- 5 + 3-^-'] ' = x-^D{_2x^{x + \)D - 2J]. 137. If, in a group of terms of the kind considered in Art. 134, m be the least value of r, and q — m be the constant value of i" — r, the group may be written A^(^o + A,xD 4-* A^oc'D^ + . . .)Z>«% . . . (i) where A^, A^, . . . , are constant coefficients, and g may be negative or fractional. Using d; as in Art. 123, to denote the operator xD, the expression in parenthesis may be reduced to the form /(^), and the group to the form x9/{{^)D^y (2) § XIL] SYMBOLIC TREATMENT OF EXACT EQUATIONS. \\l It is shown in Art. 134 that, if m is not zero, and q is zero or a positive integer less than m, every term in the expression (i), and hence the whole expression (2), is an exact derivative. The symbolic transformation expressing the result, in this case, may be effected by means of the formula deduced below. 138. We have, by differentiation, ax dx dx^ dx or D&y = &Dy + Dy ; whence symbolically - i) (i) Operating successively with ^ upon both members, we derive ^^D = &D{& - i) = D{<> - i)% &W = ^D{& - l)^ = D{d^ - 1)3; and, in general, Now, since f{&) consists of terms of the form A&'^j it follows that /{&)D = D/{{> - I).* (2) * The formula by which the homogeneous linear expression is reduced to the form /(■&)y is readily deduced from this formula. For equation (i) may be written xD^y = D[& — i)y; and, multiplying by x, x^D^y = d{^ - i)y. Changing the operand y to Dy, and using equation (2), x^'D^y = ^(& — i)Dy = D(^ — i)(i? _ 2)y. Multiplying again by x, x^D^y - ^{^ - !){■&- 2)y', and in like manner, we prove, in general, xrDry =z ^{^ - l){^ - 2) . . . {^ — r + l)y. 142 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. 1 38. Again, operating with each member of this equation upon D (which is equivalent to changing the operand from y to Dy)^ /{0)D^ = D/{0 - i)D = D'f{{^ - 2). In like manner, f{0)D^ = D^f{(^ - 2)D = D^/{& - 3); and in general, /{{y)D'» = D-'/{(^ - m) (3) 139" If ^ is a positive integer less than ;;/, we can, by this formula, write whence A^/(^)Z)«' = ^{<^ - i) ...({>- ^ + i)/(0^ - q)D"'-'!, in which the expression for the group is reduced to the same form as when q =z o. We may now remove one or more of the factors of D"'-^ to the extreme left of the symbol, thus effecting one or more, up to w — ^, direct integrations, under the condition t/iat 711 is not zero, and that q has one of the values O, I, 2 ... in — I. The equation giving the result of m — g integrations is x^/{^)n'" = Z>'"-'^{{r - m + q) ...{{> - m + i)/(i'> - m). 140. In every other case, the possibility of resolving the operator into factors of the required form depends upon the presence of a proper factor in /(i>). To show this, we have, by differentiation, Dx'J^'y = x'f^^Dy + (^ 4- i)oc^y', whence, using Dx^^' as a symbol of operation, x'^{{^ + q + i) = DxW, (2) we shall have x^f{{^)D^ = Dx<^^'(j){{>)D^ (3) We have thus a second condition * of direct integrability, and an expression for the result of integration. 141. If the first member of a differential equation be expressed in terms of the form x^f{d)D''y, the conditions given in Arts. 139 and 140 serve to show at once whether the equation can be made exact by multiplication by a power of X. For example, equation (i) of Art. 135, when written in the form considered, is x\2& + ^)Dy + (2^ + 3)(^ - 1)7 = X. The first term becomes exact, in accordance with the first condition, when multiplied by x-"" ; and the presence of the factor {& — i) shows that the second term is also made exact by the same factor. Hence, by equation (3), Art. 138, and equation (i), Art. 140, the symbolic operator may be written x^Di{2Q- + 5) + x-\2x^ + 3)]. * This condition might be made to include that of the preceding article ; for we might first, by means of equation (3), Art. 138, make the transformation xgf{^)D^ = xg-^t x»tDfnf[-Q _ m)y and then the expression for x^D^t, in terms of 1?, which is ^{■^ - \) ...{■&- m J^ I), would, under the previous condition, contain the factor i9 + ^ — w + if, which, in accordance with equation (i), should accompany x'^~^^. But, since under no other condition would this happen, and since the factor would not appear m/()? — m) unless i5 4. ^ 4. I had been a factor of /(»9), this transformation is clearly un- necessary. 144 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. I41. Again, both terms of the last factor fulfil the condition of Art. 140 when multiplied by x^y and the expression becomes 2x'Dx-\D{x^ ^ x\). The value of y obtained by performing upon X the inverse operations in the proper order is I f AXdx xl H- x^ J 2^" in which each integral sign implies an arbitrary constant. The expression is readily identified with that given in Art. 135. It will be noticed that whenever an equation becomes exact when multiplied by either of two different powers of x^ it is also susceptible of two successive direct integrations. Examples XII. Solve the following differential equations : — y — c^x -^ ^{i — x')(c2 — sin-'^), or y =: C,X -h V^(^ - 1)\C, - l0g[^ + V^C^ - l)]i. § XII.] EXAMPLES. 145 4. — ^ -f cos X -^ — 2 sin ^ ^^ — y cos x = sin 2x. dx^ dx^ dx sm^ — I 4:2 %^» = ^-pl + -p^$ + 6. ;cH^ + 2)^-^ + x{x + 3)^ - 3^ = ^, ^:r^ dx (f^)% = J(7^i(^.+|f^-)^- + ^i / d'^y , dy , 7- y^-li + 2;t:-f^ + 3;; = ^, ^2^ dx :uii = ..M + .,J.a| + .,. 8. (a^ - x)"^ + 2(2;^: + i)^ + 2;; = o, y = ^,(4:^3 _ 2^^ - f:^ - ^) + x^(x — i)\c2 — 4c, log ^ |. 10. Find three independent first integrals of the equation /" = X. f = yC^ + ^" ^f - / = \xXdx + r„ 146 LINEAR EQUATIONS: VARIABLE COEFFICIENTS. [Art. I4I. 11. Derive (a) the complete integral oi y'" = X from the above first integrals, and ()3) the integral of /^ = A" in like manner. (a), 2y = oA^dx - 2x\xXdx -f \x^Xdx + C.F. (y8), dy = xAxdx - ixAxXdx + 3^: Lr^X/jc - \x^Xdx -\- C.F. 12. Solve the equation dx^ dx (a), as an equation of the first order for _/; (^), as an exact equation when multiplied by a proper power of x. (a), y=.A+ /^^ + \{2^x + i)-3[(^\^_±^V^.r^.^. ^'^^^ -^ ^(2VAr + I)^ (2v'^,+ i)^J X' ,^y .-/ 1 3y-ir Show that the .equation {2x^ + 6^:^)/^ 4- (13^3 4- 41^^)/" + (11:^2 4- 54^^)/' — (ioa: — Sx})y' — 2;; =" -X" ipay.be written -' {& + iy{2& Jt^ i){& - 2)y -/^- =- \ + ^i(3^ 4. i)(2i'> 4- 3)(^ + 2)Dy='X, and find its integraL § XIIL] LINEAR EQUATION OF THE SECOND ORDER. I47 XIII. TJie Linear Equation of the Second Order. 142. No general solution of the linear differential equation with variab le coefficients exists when the order is higher than the first : there are, however, some considerations relating chiefly to equations of the second order which enable us to find the integral in particular cases, and to these we now proceed. If a particular integral of the equation %^-pi^Qy = o,. .... . (.) in which P and Q are functions of x, be known, the complete integral, not only of this equation, but of the more general equation can be found. For let y^ be the known integral of (i), and assume y = y,v in equation (2). Substituting, we have, for the determination of the new variable v, d^v , dy, dv , d^y, 1 dx^ dx dx dx^ dx dx + Qy.v J = X .... (3) The coefficient of v in this equation vanishes by virtue of tHe hypothesis that 7, satisfies equation (i) ; thus the equation 148 LINEAR EQUATION OF THE SECOND ORDER. [Art. I42. -. — » . . — —^ ■ *■ ■ - .. becomes a linear equation of the first order for — • or 1/. Hence dx V may be determined ; and then y ^ y^ Wdx -f C,y, is the integral of equation (2), the other constant of integration being involved in the expression for v' . 143. As an illustration, let the given equation be (I - :e)g + ^1 - J- = x{^ - ^)5, in which, if the second member were zero, y r=i x would obviously be a particular integral. Hence, assuming y =■ xv, and substituting, d' — + ( 2 + )— ±= ^(i — ^)», x^ \ I — x^Jdx or Solving this equation, we have or dx x^ and, integrating, V = "J(i - x^)^ + ^.[sin-';c -{- ~^^ J +^f § XIII.] A PARTICULAR INTEGRAL KNOWN. 1 49 Hence y r= —\x{i — x^)^ + ^[^sin-^jc + (i — x^)^^ + C2X. 144. The simplification resulting from the substitution jj/ = j/^v is due to the manner in which the constants enter the value of j/ in the complete integral. For we know that ^ is of the form y ^ ^i>'i + ^zyz + y, where f^ and j/2 are independent particular integrals of the equation when the second member is zero, and F is a particular integral when the second member is X. Hence the form of v is y2 y and that of if is -'.fey*©' so that the equation determining v' must be a linear equation of the first order. In like manner, whatever be the degree of a linear equation, if a particular integral when the second member is zero be known, the order of the equation may be depressed by unity. Expression for the Complete Integral in Terms of y^. 145. The general equation for v'y where y in the equation S+^l + e-=^ <'> is put equal to y^v, and y^ satisfies 150 LINEAR EQUATION OF THE SECOND ORDER. [Art. I45. is [equation (3), Art. 142] dx Solving this linear equation of the first order, we have , \Pdx , [ \Pdx^j , and, since j = jy^v = j, vdx, -\Pdx. , . -\pdx e J I \pdxy,j , . . \e ^ y ^ y {'—^iyJ'''^Xdx^ + c,y, + c,y,{'——dx (3) is the complete integral of equation (i) if j, is an integral of equation (2). Owing to the constants of integration implied in the integrals, the first term is, in reality, an expression for the complete integral : but the last two terms give a separate expression for the complementary function ; that is to say, for the complete integral of equation (2). 146. Thus the complete integral of equation (2) may be written y = ^ijVi + ^2>'2 where \Pdx J2 = ^,1^ —dx (4) e J This expression may, in fact, represent any integral of equation (2) ; but, when the simplest values of the integrals involved in it are taken, it gives, when y\ is known, the simplest independent integral ; that is to say, the simplest integral which is not a mere multiple of ^,. § XIII.] RELATION BETWEEN THE TWO INTEGRALS. I5I For example, in the equation .d^y dy , dx^ dx assuming, as in Art. 122, j/ = x^^ we have . w^ — 4^ + 4 = o. A case of equal roots arising, this gives but one integral of the simple form y = x"^, namely, y^ = x^. Now, in the given equation, P = —-; hence ^-j^^-^ = ;i:3 ; and, substituting in equation (4), we have ^2 = x^\ — dx = ^c^log:^ jx* for the simplest independent integral. 147. The relation between the two independent integrals y, and j/a may be put in a more symmetrical form. For equation (4), Art. 146, may be written ^^-; (I) whence, differentiating, we obtain dy2 dy, -\pdx , . This is a perfectly general relation between any two independent particular integrals of ^ + piv + e;- = O, dx^ dx 152 LINEAR EQUATION OF THE SECOND ORDER. [Art. 1 47. but it must be recollected that the value of the constant implied in the second member depends upon the form of the particular integrals y^ and y^. For this reason, the relation is better written It will be noticed that, in this equation, the change of y^ to rny^ multiplies A by w, but the change of y^ to j^ -I- my^ does not affect A. 148. We may also, by introducing jj, obtain a more symmetrical expression for the particular integral of the equation than that given in Art. 145. For, since by equation (i), Art. 147, dx = d-^, the particular integral in equation (3), Art. 145, may be written which, by integration by parts, becomes y= y\y.e\''''''Xdx - y\y,y''''Xdx, in which \Pdxy in the exponential, is to be so taken as to satisfy equation (2) ; otherwise, the second member should be divided by the constant A defined by equation (3) of the preceding article. § XIII.] RESOLUTION OF THE OPERATOR INTO FACTORS. 1 53 Resolution of the Operator into Factors. 149. We have seen, in Art. 129, that, when the symbolic operator of a linear equation whose second member is zero is resolved into factors, the factor nearest the operand y gives, at once, an integral of the equation. Conversely, when an integral is known, the corresponding factor may be inferred ; and, if the equation is of the second order, the other factor is found without difficulty. For example, in the equation ^^ ~ "^^^ - (9 - 4^);^ 4- (6 - 2>x)y = o, the fact that the sum of the coefficients is zero shows that e* is an integral. The corresponding symbolic factor is /> — i, and accordingly the equation can be written [(3 - x)D - (6 - 3^)](^ - ^)y = o. The solution may now be completed as in Art. 130; thus, putting V z=z {D — \)y^ we have ^ ^ 3^ - 6 ^^^ V X — z the integral of which is V = CeT>^{x — 3)3. Finally, solving the linear equation {D - i)y = Cei-{x - 3)3, we have the complete integral y — Ae^ + Be^^{^\x^ — 42^^ + 150^ — 183), in which B is put for the constant \C. 154 LINEAR EQUATION OF THE SECOND ORDER. [Art. 150. 150. In general, if y^ denotes the known integral, and D — y] is the corresponding factor, (Z? - ri)y, = 0, or ^ - r;;;. = o ; whence . = -!-f- . (I) y^ dx Now, in the case of the equation of the second order {D^ ^ PD ^Q)y^o, (2) the other factor must be Z> + P + 77 in order to make the first two terms of the expansion identical with those of equation (2) ; thus we have {D^P ^'ri){D-'ri)y^o; (3) which, when expanded, is D^y + PDy - (^^ + Pr^ + Ay = 0. . . . (4) T/ie Related Equation of the First Order. 151. If, regarding t] as an unknown function, we attempt to determine it by equating the coefficients of y in equations (2) and (4) of the preceding article, the result is £ + >y^ + A + <2 = o (I) Hence, to any solution of this equation of the first order, there corresponds a solution of ^ + /'^ + e> = o ' . . (.) dx^ dx §XIII.] RELATED EQUATION OF THE FIRST ORDER. 1 55 Equation (i) is, in fact, merely the transformation of this equa- tion when we put, as in the preceding article, . . . . : ' = 51 <3) Although of the first order, equation (i) is not so simple as equation (2), which has the advantage of being linear. In fact, the transformation just mentioned is advantageously employed in the solution of an equation of the form (i). See Art. 193. Since the complete integral of equation (2) is of the form y = c,X,^c,X, (4) where X^^ and X^ are functions of x, that of equation (i) is of the form ^ c,X^-\-c,X, X, + cX,' • • • • V5; which indicates the manner in which the arbitrary constant c enters the solution. The particular integrals of (i) produced by giving different values to c correspond to independent integrals of equation (2), that is to say, integrals in which the ratio c^ : c^ has different values ; the integrals in which c =■ o and ^ = 00 in the expres- sion (5) corresponding to the integrals X^ and X^ of equation (2). The Transformation y = v/{x). 152. If, in Art. 142, we replace j, by ze/„ an arbitrary function of x, the result is that the equation S + ^^ + e. = x (0 dx^ dx 156 LINEAR EQUATION OF THE SECOND ORDER. [Art. 1 5 2. is transformed, by the substitution y = o'.z', (2) into d^v I n dv , ^ XT / \ ^ + ^■^+ ^■'' = ^" <3> where p.=^^p + p, (4) w^ dx c. = -i-!^' + ^^ + a (5) a/, dx^ Wi dx ^. = - (6) Pii Qiy and X^ are here known functions of x ; thus the equation remains linear when a transformation of the dependent variable of the form y =■ vf{x) is made. 153. The arbitrary function w^ can be so taken as to give to /*! any desired value ; thus, if /*, is a given function of Xy we have, from equation (4), ^ = :^{P,^P)dx', whence Z£/, == ^ J ^ J • . . (7) Substituting in equations (5) and (6), we find, for the values of Ci and X^y in terms of P„ ^■=e--i(^-'-^) + ' = o. dx^ dx Here P = — 2 tan x ; therefore, by equations (i) and (4), Art. 155, I tan X dx «/o = = ^' y z=. c,x -\- C2x\e^~ + I. XIIL] EXAMPLES. 163 3- S-| + (-')- = ^' 2.eAe^ ^^dx •\- eAe"^ ^^\e 2 "*" y = c^e^ + ; = c^{xi^ — a) + c^x. dx^ dx , d^y dH dy , 6. — ^ — ^— ^ -^ xy = o, dx^ dx' dx ^ y = c^e^ + ^2^--^ 4- r. /zjc^ A' ^^t: x^ JeAe^^^-^dx — ^-•rU^-*'' + ^^^Y j; = c^x-'e*"^ + f2j«:-2^-»«^. ^2^J^' . ^^ _ 2^^ 4- 2V - o ^ = r.jr^ + c ,x + (r3(::\;Mji;-3^-^^A: — xXx-^e-'^dxy 9. (2^3 _ «)^ — 6x^-^ + 6^y = o, y = ' + P'^-f + Qy = O, dx dx dx^ dx by eliminating Q and nitegrating the result. § XIII.] EXAMPLES. 165 22. Find the symbolic resolution of D^ corresponding to the integral x of the equation D^y — o. ^=(^^9(^-9- 23. Find the symbolic resolution oi D^ — i corresponding to the integrals cosh x and sinh x of the equation (Z>^ — \)y = o. D^ — I = (jD + tanh x) (D — tanh x) = (Z> + coth:v)(Z> - coth:^). 24. Show that the ratio s of two independent integrals of dx^ ax satisfies the differential equation of the third order where Qo is the function defined in Art. 155. 25. Show that, if P be expressed in terms of 0, the equation of Art. 160 may be written H'2 + Qy = X. 26. Prove that, in the equation £Z + P|' + e;- = o, dx^ dx the function is an invariant with respect to the transformation z = ^{pc)^ 1 66 SOLUTIONS IN SERIES. [Art. 1 6 1. CHAPTER VII. SOLUTIONS IN SERIES. XIV. Development of the Integral of a Differential Equation in Series, i6i. In many cases, the only solution of a given differential equation obtainable is in the form of a development of the dependent variable y, in the form of an infinite series involving powers of the independent variable x. Moreover, such a development may be desired, even when the relation between X and y is otherwise expressible. If we assume the series to proceed by integral powers of x, an obvious method by which successive terms could generally be found is as follows. Sup- posing the equation to be of the ;?th order, and assuming, for the 11 arbitrary constants, the initial values corresponding to X = o oi y and its derivatives, up to and inclusive of the {n — i)th, the differential equation serves to determine the value of -^ when x =z o. Differentiating the given equation, ax** we have an equation containing =^, which, in like manner, dx" + ' serves to determine its value when ;ir = o, and so on. Thus, writing out the value of y in accordance with Maclaurin's theorem, we have the values of the successive coefficients in terms of n arbitrary constants. § XIV.] LINEAR EQUATIONS. 167 162. It would usually be impossible to obtain, in the manner described above, the general term of the series. We shall therefore consider only the case of the linear equation (and such as can be reduced to a linear form), in which case we have a method, now to be explained, which allows us to assume the series in a more general form, and, at the same time, enables us to find the law of formation of the successive coefficients. Since we know the form of the complete integral of a linear equation to be y = c,y, + ^2^2 + . . . + Cnyn + y, our problem now is the more definite one of developing in series the independent integrals j^„ jz • • • JF«, of the equation when the second member is zero, and the particular integral Y of the equation when the second member is a function of x. No arbitrary constants, it will be noticed, will now occur in the coefficients of the required series, except the single arbitrary constant factor in the case of each independent integral. Development of the Independent Integi-als of a Linear Equation whose Second Member is Zero, 163. We have seen, in Art. 122, that if, in the first member of a homogeneous linear equation whose second member is zero, we put y = Ax^\ the result is an expression containing a single power of ;ir ; so that, by putting the coefficient of this power equal to zero, we have an equation for determining m in such a manner that y = Ax'*' satisfies the differential equation, A being an arbitrary constant. If we make the same substitution in any linear equation whose coefficients are rational algebraic functions of Xy the result will contain several powers of x. Let us, for the present, suppose that it contains two powers of x^ and also 1 68 SOLUTIONS IN SERIES. [Art. 1 63. that the differential equation is of the second order. The term containing — ^ in the differential equation will produce at least one term, in the result of substitution, involving m in the second degree ; hence at least one of the coefficients of the two powers of x will be of the second degree in m. Let X"*' and ;ir'«'-^^ where s may have any value, positive or negative, be the two powers of ;r, and let the coefficient of x*"' be of the second degree. Now let m be so determined that the coefficient of X'"' shall vanish, and suppose the quadratic equation for this purpose to have real roots. Selecting either of the two values of nit the coefficient of x^'+^ will, of course, not in general vanish. Suppose, now, that we put for y, in the first member of the differential equation, the expression AoX^" + A^x*"-*-^, the result will contain, in addition to the previous result, a new binomial containing ^„ and involving the powers ;ir''''+^and x*"'+''^; the entire coefficient of x""'-^^ will now contain A^ and A,, and may be made to vanish by properly determining the ratio of the assumed constants A^ and A^. In like manner, if we assume for J/ the infinite series y = AoX^^ + AiX'*' + ^ + ^a^'^H-" H- . . . , or y = t^ArX^' + ^^, we can successively cause the coefficients in the result of substitution to vanish by properly determining the ratio of consecutive coefficients in the assumed series. If the series thus obtained is convergent, it defines an integral of the given equation ; and, since in the case supposed there were two values of jn determined, we have, in general, two integrals. If s be positive, the series will proceed by ascending powers, and, if s be negative, by descending powers, of x. § XIV.] DETERMINATION OF THE COEFFICIENTS. 1 69 \ 164. For example, let the given equation be ^ - "i - ^-^ " ° <■> The result of putting A^x^ for y in the first member is m{in — 1)^0^"^ -2 — (m -{- p)AoX'^ (2) The first term, which is of the second degree with respect to M, will vanish if we put m{m — i)Ao = o (3) The exponent of x in this term, or m', is m -- 2, and the other exponent, or in' + s, is m ; whence s = 2. We therefore assume the ascending series y = 2o^r^^' + 2'-, and, substituting in equation (i), we have 2^5 (w + 2r){m -\- 2r — i)ArX"' + ^''-'' — {in + 2r -\- p)ArX'^ + ^^\ = o, (4) in which r has all integral values from o to co. In this equation, the coefficient of each power of x must vanish ; hence, equating to zero, the coefficient of x^ + ^^-^y we have {jH + 2r){i7t + 2r — i)Ar — {m -\- 2r — 2 -{- p)Ar^i = o. (5) When r =: o, this reduces to equation (3) and gives m — o or m = I ; and when r > o, it may be written {m 4- 2r){m + 2r — i) which expresses the relation between any two consecutive coefficients. I/O SOLUTIONS IN SERIES. [Art. 1 64. When m =-0y this relation becomes _ / + 2r- 2 . . 2r(2r — i) whence, giving to r the successive values i, 2, 3 . . ., we have ^^ - 3.4 ^^ - 4 ! ^°' . _ / + 4 . _ /(/+ 2)(/ + 4) . The resulting value of y is y = AJi + /^H-/(/ + 2)^ L 2 ! 4 ! + /(/+ 2)(/ + 4)f^ + ...]. (7) Again, giving to m its other value i, the relation (6) between consecutive coefficients becomes p + 2i {2r + i)2r jj / + 2r — T D whence / + 3 „ (/+!)(/ + 3) Tr^^ = s Bz = — 7-;:— Bi = —^ Bo ^3 = -^- ^2 = ^-j Bo; and the resulting value of j/ is ^ = 5„ 1^* + (/ + i)|i + 0> + i)(/ + 3)|! + . , .J. (8) § XIV.] CONVERGENCY OF THE SERIES. 17I % . Denoting the series in equations (7) and (8), both of which are converging for all values of Xy by y^ and /a, the complete integral of equation (i) is y = Aoy^ + Boyz (9) 165. It will be noticed that the rule which requires us to take, for the determination of m, that term of the expression (2) which is of the second degree in m was necessary to enable us to obtain two independent integrals. But there is a more important reason for the rule ; for, if we* disregard it, we obtain a divergent series. For example, in the present instance, if we employ the other term of expression (2), Art. 164, thus obtaining m = —p and i- = — 2, the resulting series is ' , , /(/+ !)(/ + 2)(/+ 3) , 1 2.4 J The ratio of the {r + i)th to the rth term is _ {p+ 2r- 2){p+ 2r- I) _ X- 2r and this expression increases without limit as r increases, whatever be the value of x. Hence the series ultimately diverges for all values of x. When both terms in the expression corresponding to (2) are of the second degree in m, we can obtain two series in descend- ing powers of x as well as two in ascending powers ; and, in such cases, the descending series will be convergent for values of X greater than unity, and the ascending series will be con- vergent for values less than unity. lyi SOLUTIONS IN SERIES. [Art. 1 66. The Particular Integral. i66. When the second member of a linear equation is a power of Xy the method explained in the preceding articles serves to determine the complementary function, and the particular integral may be found by a similar process. Thus, if the equation is S - "I - y*-^ = ^*' the complementary function is the value of y found in Art. 164, To obtain the particular integral, we assume for y the same form of series as before, and the result of substitution is the same as equation (4), Art. 164, except that the second member is x^ instead of zero. Equation (5) thus remains unaltered, while, in place of equation (3), we have m{m — i)AoX»^-^ = x^. This equation requires us to put w — 2 = i, and m(m — i)Ao = i j whence /// = S, and Ao = r%. The relation (6) between consecutive coefficients now becomes Ar = / + 2r 4- ^ (2r + |)(2;'4-|) '-" or . _ 2(2/ 4- 4r + i) yf hence 2(2/ + 5) '*• ~ 7.9 ^°' _ 2(2/ + 9) , _ 2'(2;> + 5)(2/ + 9) '^' - II.I3 ■ ~ 7:9^11:^3 °' § XIV.] BINOMIAL AND POLYNOMIAL EQUATIONS. 1/3 and the particular integral is i--i.j,t'"^-^"^+''"^-"""^*'"«-+. ^ 7-9 7-9-II-I3 If the second member contained two or more terms, each of them would give rise to a series, and the sum of these series would constitute the particular integral. Binomial and Polynomial Equations. 167. If we group together the terms of a linear equation whose coefficients are rational algebraic functions of x in the manner explained in Art. 134, we can, by multiplying by a power of Xy and employing the notation ;ir— - = ^, put the equation in ax the form /xWj^' + ^VaWj + ^V3W;' + ... = o, . . (i) in which s^, s^, . . . are all positive, or, if we choose, all negative. The result of putting A^x"*' for y in the first member is AoMm)x»' + AJ:,{m)x**'^'^ + AoMm)x'>' + '^ -h . • • . (2) Equations may be classified as binomial^ trinomial, etc., accord- ing to the number of terms they contain, when written in the form (i), or, what is the same thing, the number of terms in the result of substitution (2). Thus, the equation solved in Art. 164 is a binomial equation. In the general case, the process of solving in series is similar to that employed in Art. 164, the form which it is neces- sary to assume for the series being J^; == t^ArX^'^-^^, where s is the greatest number, integral or fractional, which is contained a whole number of times in each of the quantities i-„ jj, etc. As before, m is taken to be a root of the equation 174 SOLUTIONS IN SERIES. [Art. 167. /,(;«) z= o, and Aq is arbitrary ; but, when the coefficient of the general term in the complete result of substitution is equated to zero, the relation found between the assumed coefficients ^01 -^i» -^a, etc., involves three or more of them, so that each is expressed in terms of two or more of the preceding ones. We can thus determine as many successive coefficients as we please, but cannot usually express the general term of the series. We shall, in what follows, confine our attention to binomial equations of the second order. Finite Solutions. 168. It sometimes happens that the series obtained as the solution of a binomial equation terminates by reason of the occurrence of the factor zero in the numerator of one of the coefficients, so that we have a finite solution of the equa- tion. For example, let the given equation be dll^a^l- 21.^0 (I) dx' dx x" ^ ' This is obviously a binomial equation in which s =. \ \ hence, putting 7 = XArX^^r^ we have ^\\_{fn ■\- r){m -\- r ^ i) — 2\ArX'^ + r-2 + a{m -f- r)ArX'^^->'-'^\ = o. Equating to zero the coefficient of ;i:'« + ''-2, we have {ni j^ r -\- \){m ■\- r — 2)Ar + a{m -\- r — i)^r_i = o, which, when r = o, gives (« + i)(/« — 2)^0 = o; (2) § XIV.] FINITE SOLUTIONS. 1 75 and, when r> o, The roots of equation (2) are m =z —i and m =^ 2; taking m = —ij the relation (3) becomes Ar = -a ^ ~ ^ Ar.ry (4) r{.r - 3) in which, putting r = i, and r = 2, we have A^ = —a ~^ Ao, l(-2) A^ = —a ^ A, = o. 2(-l) All the following coefficients may now be taken equal to zero,* * In general, when one of the coefficients vanishes, the subsequent coefficients in the assumed series I>o Arx»i + rs must vanish ; in other words, the value of y can contain no other terms whose exponents are of the form m + rs. But, in the present case, the assumed form \s y =. 'Lo Arxr— i ; and this includes the powers x^, x^ . . . which we know to be of possible occurrence since the other value of m in this case is 2. Accordingly, if we continue the series, it recommences with the term containing x'^. Thus, putting r = 3 in equation (4), we obtain ^3 = -al-A2 = ?, 3.0 o which is indeterminate ; then, putting r = 4, 5, etc., we have A 4 = —a — A3, As = —a -^ A4 z= a^^A^, etc. 4.1 5.2 4.5 Thus, the assumed form j/ = 2© Arx^—r really includes, in this case, the complete integral y = aJI -^) + A^'(i - ^ax + 1-a^x^ ±-a^x^ +'\ \x 2/ \ 4 4.5 45-6 / 176 SOLUTIONS IN SERIES. [Art. 1 68. SO that we have the finite solution * ^.;-. = ^^-(i - 2.) = ^..(i - 5). 169. For the other solution, taking w = 2, the relation (3) becomes whence Br = —a-. ; —Br-i\ 1.4 B^ = -al^B. = a^l^B.. B, = -aA^B, = - a^-^B, 3.6 4-5-6 Hence Boy^ = BoX^fi - -ax + -^a'x' ^a^x^ + . . -V \ 4 4.5 4.5-6 / and the complete integral is 2 — ax / 2 z \ y = Ao h BoxH I ax + -^a^x^ — . . . ) . -^ ° 2^ ^ °" V 4 4-5 / 170. Since we have, in this case, a finite integral of a linear equation of the second order, namely, * In like manner, if, in a trinomial equation, the coefficients between which the relation exists are consecutive, a finite solution will occur when two consecutive coefficients vanish. § XIV.] EXAMPLES. 177 equation (4), Art. 146, gives the independent integral 2X J (2 — ^.::ic)^ We must therefore have y^ = Ay^ -}- ^j/j where 7, and y^ are the integrals found in the preceding articles, and the constants A and B have particular values to be determined. Since both J/'/ and j/2 vanish when ;i; = o, while y^ does not, we shall have A = o; and, comparing the lowest terms of the development of the integral with the series J2, we find B = ^; hence 2 — axt x^e-^^ 7 x^r 2 I 3 2 2 "1 ax = — I ax + -^-a^x^ — .... X J (2 - axy 61 4 4.5 J Examples XIV. Integrate in series the following differential equations : — I. x—^ + {x + «)-/ 4- (« + i)y = o, ax^ ax y = Afn — {n -\- i)x -\- {n + 2)—^ — (« + 3)— + . . . j 4- Bx'-^(l + —^— X 4- r^^ r X' \ n — 2 (« — 2)(/2 — 3) (« - 2)(« - 3)(;z - 4) ^3 4- ■> 2. ^ 4- ^J' = o, \ 3i 6! 9! y 4. b(x _ £^^4 4. ^:x7 - . . A 178 SOLUTIONS IN SERIES. [Art. I/O. ^ dx" dx ^ ^^ y = Ax( I H H h \ 2.5 2.4.5.9 2.4.6.5.9.13 / \ 2.3 2.4.3.7 2.4.6.3.7.1 1 / ^ x^ 1.3 1.3.3.7 1.3.5.3.7.11 4. ^0+^£ + ^'^>' = 2, 4. ^/l _ 2-2-^3^3 4. 2^a^x^ - 2^^a9x'^ + . . .Y \ 5! 8! 11! J ^ dx^ -^ * y — A{\ f- ■ 1- ... I \ 3-4 34-7-8 3.4.7-8.11. 12 / + Wi - ^V -^!^ ^^^" + . . .\ \ 4.5 4.5-8.9 4-5-8-9-12. 13 / , jr*/ ax^ , a^x^ \ , x^f ax^ , a^x^ \ 2\ 5.6 5.6.9.10 / 6\ 6.7 6.7.10.11 / 6. X — ^ H- (^ + «)— + (^ — i))* = ^*-'', ^ n i! ;z + i2! ;2 4-23! / 4. J^lZlf^ L_ £ + I ^ - . . .V 2 - n\ S — n 2 (3 — ^) (4 — «) 3 / § XIV.] EXAMPLES. 179 7- ^^ + 4^ + ^ = °. dx^ dx y V 3i 5I 7! 9 J / 8. ^^+(^ + 2^^)^-4J = o, \ 5 5-6 5-6-7 / \^ 3^ 3/ 9. (a: - ^)-^ + 3^^ + 2;; = o, y = A(6 - 4X + x^) + ^i-i:A£, Show also that x-^(i — ic)'» is an integral. 10. (4^3 _ 14^ _ 2;^)^ - (6J1-2 _ 7^ + i)^ ^2 ^^j; H- (6^ — i)y = o, J = Axi(i H- 2a;) + B(i — x). 11. ^2^ + ^2^ ^(x- 2)y = o, dx^ dx ^ \3 41 52! 63! / 12. Denoting the integral in Ex. 11 by Ay^ + ^j^a, find, by the method of Art. 146, an independent integral, and express the relation between the integrals. , /2 , , \ & J2 = e-^(- ^ 2 -\- x\ = 2y, - y^. y = Ax^fi + ^ -f -^ + -^ + . . A + ^/l + I + £\ \ 4 4.5 4.5-6 / \^ 2/ Show also that x-^e^ is an integral. l80 SOLUTIONS IN SERIES. [Art. I/O. 14. ^Ui - 4A)yi + [(I - n)x - (6 - 4«)^]^ + «(l — «)^J = o, A J X i ^(^ + 3) 3 «(« + 4)(« + 5)- , , \ y = ^:r«( I + «a: H j^ ^^ H — j ^3 + . . . j ^ j,f , ^(^ - 3) 3 ;2(;2 - 4)(« - 5) , , \ + JB\^i - nx + -^ X' : —^ ^3 + . . .j. y = Ax-^li - -^ + — W ^^^/'i - -^x + -4:5_^2 _ . . .\ \ 5 20/ V 1-7 1-2.7.8 I 16. (a^ + :v^)— ^ + ::c-=^ — n^y = o, ^ \ 2\a^ 4! «4^ y 17. Denoting the integral given in Ex. 16 by Ay, + By2, show and find the corresponding result when « = o. log [x + v/(^^ + ^^)] = log^ + ^-i-^ + ^ — -... a 2 3^3 2.4 5^5 18. Expand sin(^sin-' jc) and cos («cos-':r) by means of the differential equation ,3 , (' - "^>5^ - "i + '^'^ = °' of which they are independent integrals, sin (« sin- ' ^) = axil 1— x^ + ^^ —^ ^ x^ ^ , , .h cos(tf sin- ^jc) = I - ^x^ + flifi-ZUtl^ - . . . 2 1 4 ! § XV.] CASE OF EQUAL VALUES OF m. l8l XV. Case of Equal Values of m, 171. If the two roots of the equation determining m are equal, we can determine one integral of the form y = ^ArX'" + ^^ by the process given in the foregoing articles ; but there is no other integral of this form. We therefore require an independ- ent integral of some other form. For example, let the given equation be ^^' "" ""'^S "^ ^' ~ ^^^^fx - ^->' = °> • • • (') a binomial equation, in which we may take s = 2, or s = —2. Assuming ^«, ^ we have, by substitution, 2~[(;/2 + 2ryArX»' + ^^--' — {m + 2r -\- iyArX^' + ^^+^2 = o. Equating to zero the coefficient of z"' + ''''-^, we have (m -f 2ryAr — (m -\- 2r — lYAr-x = 0.. . . (2) Putting r =. Oy m^Ao = o ; whence m = o, the two values of m being identical. Putting 7^ = o in equation (2), the relation between consecutive coefficients is _ {2r- lY , . ^''" {2ry ^"-^^ whence we find the integral (-.2 1-2 ,2 -2 ,2 ^2 \ 2^ 2^4^ 2^.4^.6^ / 1 82 SOLUTIONS IN SERIES. [Art. 1 72. 172. To obtain a new integral, we shall first suppose the given equation to be so modified that one of the equal factors in the first term of equation (2) is changed to w + 2r — //, so that one of the values of m becomes equal to //, while the other value remains equal to zero. We shall then obtain the complete integral of the modified equation, in which, after some trans- formation, we shall put h = o, and thus obtain the complete integral of equation (i). The altered relation between consecutive coefificients may be written {m -\- 2r){m -\- 2r — h) in which, for a reason which will presently be explained, // is put in the place of h. Hence, when tn = o, we have and the first integral now is >'•='+ -T^-T^^' + —f T^TT FT^' + • • • • (5) 2{2 — h) 2.4(2 — /? )(4 — ^ ) Putting m =^ km equation (4), we have B = (2r - I + hy jg {2r + h){2r - h' + h) ''"'' and the second integral is ^' \ (2 + h){2 - h' + h) + (I + hyii + hY ^ + .,.\ (6) § XV.] CASE OF EQUAL VALUES OF m. 1 83 if, The object of introducing // in equation (4), in place of the equal quantity //, is that, when equation (6) is written in the form \\iiji) shall be such a function of h that, by equation (5), . , y,^ V'(o)- Developing y^ in powers of //, we have, since x^ = ^^^s* ^^3 = (i + h\ogx + . . .)[;'i + #'(0) + ...]; hence the complete integral is y = Aoy, + Boy. + BJi\_y,\ogx + r\,\6) + ...]; or, replacing the constants A^ + B^ and BJi by ^ and B^ y = 4);, + By, \ogx + ^i/.'(o) -}-...,... (7) in which we have retained all the terms which do not vanish with h, and, when h = o, y, resumes the value given in equation (3). 173. It remains to express y\ii^ in terms of x. In doing this, we may, since //is finally to be put equal to zero, make this substitution in the value of \\i{Jt) at once, and write ^^ ^ (2 + hy ^ (2 H- /0H4 -f- hy ^ ^ ^ Denote the coefficient of x'^^'m. this series hy Hr, so that H^^^ it and when r > o, H = (I +/^)-(3 + >^)-...(2r- I +hy . , . (2 4-^)^(4 + /^)-...(2r + y4)^ . • • vyy 1 84 SOLUTIONS IN SERIES. [Art. 1 73. then and ^{h)^X'^-^X^r^XH/-:^^X^r^. . . (10) an an JTT in which unity is taken as the lower limit because — — ° = o. ah From equation (9), d\ogHr _ 2 2 I _ , + dh \ •\- h 3 + ^ 2r— 1+^ 2 -\- h ^ + h 2r -{- h* which, when ^ = o, becomes dXogHjT] _ 2 2 2 22 2 ^'^ Jo ^ 3 2/* — I 2 4 2r whence, putting // = o in equation (10), and denoting i//'(o), when thus expressed as a series in x, by y, Hence, when // = o, equation (7) gives for the complete integral of equation (i)* y = Ay, + BiyAogx + y'), where y, and _^'are defined by equations (3) and (11). * For the complete integral when we take s — —2, see Ex. XV. 7. §XV.] INTEGRALS OF THE LOGARITHMIC FORM, 1 85 Case in which the Values of m differ by a Multiple of s. 174. When the two values of m differ by a multiple of j, the initial term of one of the series will appear as a term of the other series ; and the coefficient of this term will contain a zero factor in its denominator. Hence, unless a zero factor occurs in the numerator,* the coefficient will be infinite ; and, as in the preceding case, it is impossible to obtain two inde- pendent integrals of the form ^ArX'^^''\ For example, let the given equation be ^2(x 4. oc)p- + ^^ + (I - 2x)y = o. . . . (i) dx^ dx Putting y = A^x"^ in the first member, the result is Aoiur" + i)x''' 4- Ao{m^ — m — 2)x''' + ^. Choosing the second term as that which is to vanish by the determination of m, because the first would give imaginary roots, we have rn — —\ or m — 2, and ^ = — i ; hence, putting y — '^^A^x"'-'', lo\{m — r -\- i){m — r — 2)ArX»'-^ + '' + [(;/z — ry H- i^ArX^'-^l = o; and, equating to zero the coefficient of x'"-'^ + '', {m — r-\- i){m — r — 2) A + [(^^ — r + i)^ + i]^r-i = o. (2) * It is immaterial whether the zero factor in the numerator first occurs in the term in question, or in a preceding term ; the result is a finite solution. An example of this exceptional case has already occurred in Art. 168, where s = i, and the values of m differ by an integer. 1 86 SOLUTIONS IN SERIES. [Art. 1 74. When ;;/ = —!, the relation between consecutive coefficients is r{r + 3) and the first integral is Aoy^ = AoX-^i —x-^ H ^^^x-' \ 1.4 1.2.4.5 1.2.3.4.5.6 / Putting m = 2, the relation is and the second integral takes the form Boy2 = Box^fi 5_ ^-i + _5^ \ —2.1 — 2( — 1).1.2 5>2.i 2(— l).O.I.2.3 .), (4) in which the coefficient of x-'' is infinite. Thus, the second integral of the form ^A^x*^-^''^ fails, and we require an inde- pendent integral of some other form. 175. To obtain the new integral, we proceed as in Art. 172. Thus, supposing the second factor in the first term of equation (2) to be changed to m — r — 2 — h, so that the second value of m is now 2 + h instead of 2, and using // as in Art. 172, the relation between consecutive coefficients now is A - {m - r + lY + 1 . . . § XV.] INTEGRALS OF THE LOGARITHMIC FORM. 1 8/ When w = — I, this becomes ^(^ + 3 + ^ ) and we have Aoy^ = Aox-Ui -—^——x-' + £i5 _^-2 _ . . \ (6) i.2(4 + /^0(5 +^') / ^ Putting m =z 2 -^ k, the relation between the coefficients in jj Br= (r-^-ky + i B and the new value of B^y^ is \ (-2 - /^)(i 4- /^'- /^) y in which the first term which becomes infinite when ^ = o is ^ [(-2 - hf + I] [(-I - hf + i][(-/^r+ I] ^-1 + ;^ /_x Denoting the coefficient of this term by —, and the sum of the preceding terms in y^ by 7", we may write B^y, = B,T , B J (i -hy ^T \ If now we write this equation in the form Boy^ = BoT + ^xhxp{h), n 1 88 SOLUTIONS IN SERIES. [Art. 1 75. equation (6) shows that /, = 1/^(0) ; hence the complete integral may be written y = ^ojF. + B,T + |(i -h ^logx 4- . . .)[jx + ^'A'(o) + ...]» or, putting A for the constant ^o + y » y=. Ay, -h BoT + ^^'.log^ + ^f (o) + (9) In this equation we have retained all the terms which do not vanish with h_\ from the value of By as defined by the expres- sion (7), we see that, when h =. Oy B^B -^ = ^^0;. . . . (10) (_2)(-i). 1.2.3 6 and, when ^ = o, we have, from equation (4), T = :x^ -\- ^x + i (11) 176. The expression for i/^'(o) as a series in x, which we shall denote by j', is found exactly as in Art. 173. Putting ^ ' = o at once, in the value of il/{/i) as defined by equation (8), we have ^^ ' V (I -A)U^k) (I -/i){2 _/0(4-/0(5 -^) / and, writing this in the form we have Ho = i, and, when r> i, H = r(i - hY + iir(2 -hy + x^... \(r - hy + 11 (I - A) (2 - /i) . . . (r - /5)(4 - /5) . . . (r + 3 - /i) § XV.] INTEGRALS OF THE LOGARITHMIC FORM. ll Hence in which ^\h) = X-^X{-^yHj^-^^^^X-r, . . . (12) dXogHr ^ 2(1 - h) 2(2 - h) __ dh (1-/^)^ + 1 {2 - hy + 1 (^r ^ hy + 1 1 - h 2 - h r — h 4 — k ^ + 3 When /i = o, this becomes Y/lOg^r "] _. _ __2 4__ _ _ ^ _ ^A J 12 + 1 2^ + 1 r' + I i + I + ... + I+i + £ + ...+ ^ 12 r 4 5 ^ + 3 hence, putting /^ = o in equation (12), we have L1.4V2 I 4/ » _JJ_/? + 4 _ I ^ I _ i ^ i\^_, + . . ~ 1.2.4.5V2 51245/ (13) Now, putting ^ = o in equation (9), substituting Bo = ^B from equation (10) and the value of T from equation (11), we have, for the complete integral of equation (i), y = Ay, + Bilx" + 3^ + 3 + ;;,log^ + y'), where y^ and y' are defined by equations (3) and (13). 1 90 SOL UTIONS IN SERIES. [ Art . l^f^ Special Forms of the Particular Integral. 177. We have seen, in Art. 166, that the particular integral? when the second member of the given equation is a powel of X, may be expressed in the form of a series similar to those which constitute the complementary function. Special cases arise in which the particular integral either admits of expression as a finite series, or can only be expressed in the logarithmic form considered in the preceding articles. In illustration, let us take the equation (^--)g-| = ^-' (') of which the complementary function is A sin-^;ir + B. Putting y = 2"^r^'« + =^ we have ^Ar\^{m -f- 2r){m + 2r — i)x^^^^-'^ — (»/ + 2ryx^^^^'\ = px'^'i (2) whence, when ^ > o, {m + 2r){7n -}- 2r — i)Ar — {m -{- 2r — 2)^Ar-i = o, and the relation between consecutive coefficients is {m + 2r){m -j- 2r — i) For the complementary function, we have m = i, or m = o. Putting m =: I in equation (3), . _ (2r - i)^ . 2r{2r 4- i) whence ' + i:3^ + 5:ili-^ + ---) • • • (4) § XV.] SPECIAL FORMS OF THE PARTICULAR INTEGRAL, 191 This is the value of sin-';tr. The series corresponding to w = o reduces to a single term, so that yz = I. For the particular integral F, we have, from equation (2), whence m = a + 2, and Aq = (a + i)(« + 2) Putting m = a + 2mthe relation (3), yt __ (^ + 2ry . , {a + 2r + i){a -^ 2r + 2) hence («+ !)(«+ 2)V (d! + 3)(« + 4) + (^ + 2)-(^ + 4)- ^ + ...Y (5) (« + 3)(« + 4)(« + 5)(« + 6) / This equation gives the particular integral except when a is a negative integer ; for instance, when ^ = o, and / = 2, it gives Y = x^li +^x^ + -^^x^ + ...Y \ 34 34-5-6 / which, as will be found by comparing the finite solution of equation (i) in the case considered, is the value of (sin-';l^)^ 178. Now, in the first place, if « is a positive odd integer, all the powers of x which occur in F occur also in ji ; and, when this is the case, we can obtain a particular integral in the form of a finite series. For example, if <3: = 3, we have K = ^(i+f^ + /^^ + ...Y 4.5 V 6.7 6.7.8.9 / 192 SOLUTIONS IN SERIES. [Art. 1/8. If we write this equation in the form 2.3/ 2.3.4.5 \ 6.7 / the second member is equivalent to the series j„ equation (4), with the exception of its first two terms. Thus ^ = ..-(. + i4 or y = ^,.-^(. + I.3), and, since the first term of this expression is included in the complementary function, we have the particular integral Y = X x^, 3 9 This finite particular integral would have been found directly had we employed a series in descending powers of x. 179. In the next place, when « is a negative odd integer, the initial term of ^, will occur in Y with an infinite coefficient. Thus, if ^ = —3 in equation (5), Art. 177, the second term contains the first power of x and has an infinite coefficient. To obtain the particular integral in this case, suppose first that ^ = —3 ■\- h\ then equation (5) gives y= /^-^^^ (-2 +/^)(-i +/^) /(-I ^-hyx^^f^ I (I ^hy ^ \ Putting § XV.] SPECIAL FORMS OF THE PARTICULAR INTEGRAL. 1 93 equation (4), Art. 177, shows that j, = »/^(o) ; and we may write y = r -f- ^(i + h\ogx + . . .)[jx + h^\,\6) + . . .] .here TV is a quantity which remains finite when h =1 o. Expanding, and rejecting the term -y-Ji, which is included in the complementary function, we may now take, for the particular integral, r = T + NyAogx + iVV'(o) 4- . . . , in which we have retained all the terms which do not vanish P P with h. When h = o, the values of T and" JV are — and - 2X- 2 respectively ; and, finding the value of «/''(o), as in Arts. 173 and 176, we hav^, for the particular integral, 1^.3^ /2 2 I I I l\ "1 + ^7 + 5-i-3-4-5>+---J- 180. In like manner, when a is a, negative even integer, the term containing x°, corresponding to j/j, occurs in V with an infinite coefficient. Thus, if ^ = — 4, the second term of the series in equation (5), Art. 177, is infinite. But, putting a = — 4 + //, we have V = /-^-'"^ . /(-2 + /lY (-3 + ^){-2 + h) (-3 + A)(-2 -h /i){-i + h)k \ (i -j- h){2 -h^) / or K= T+^{i -h/i\ogx + ...)xl^{h). 194 SOLUTIONS IN SERIES. [Art. l8o. In this case, when \\i(Ji) is expanded in powers of //, the first term is unity, and there is no term containing the first power of // ; hence, rejecting the term — which is included in the complementary function, and then putting h =■ o, we have the particular integral 6x' 3 ^°^'^* Examples XV. Integrate in series the following differential equations : — ax^ ax y = (A + B\ogx)(j - ^ + ^ ^i_ + . . \ rt^v , dy \ ^ L 2^ 2^4^ \ 2/ 2^4\6^V 23/ J V 1.2 I.2^3 I.2^3^4 / ^B + ^^cf^ A _ J^U + ^U . . .1 |_1.2 1.2 I.2^3\I.2 2.3/ J §xv. EXAMPLES. 195 4. ^3^ - (2^ - i)y = o, \ 1.4 1.2.4.5 1.2.3.4.5-6 / + 3^(4^^ + 2a: + I) L1.4V 4/ 1.2.4.5 \ 245/ J ^* "^'^ "^ "^^^ "^ ^^^ "^ ^'^'^ ~ ^^-^ ^ ^' \ 3 I 32 1 335 / 3 + ^X4£/I _ I + A _ 5^/i _ i + , + iW . . 1 L3 A3 4 / 3 2 \3 5 2/ J 6. (x — x') — ^ — J = o, dx^ y = (A + B\ogx)x(i + —X + -^^' -I ^^^ ^3 + . . . ) •^ ^ ° '' \ ' 1.2 i.2''.3 1.22.32.4 ' / Ll.2\I I 2/ ^ I.22.3\l ^3 I 2 3/ 7. Find the integral of ^^(i _ ^2) J + (i - 3^')-r - ^>' = o, [equation (i), Art. 171,] when x > 1. y={A + B\ogx)x-{i + ^x-- + ^x-^ + . . .^ \ 2^ 2^.42 / — 2Bx- _2' 1.3 2^4=\l.2 3.4/ J 196 SOLUTIONS IN SERIES. [Art. 180. 8. ^ + 2f = o, (Aax^ A*a*x A^a^x^ \ [AaxUi i\ A^a^x Iy i i i\ l 9. ;e^ - (a:' + 4^) ^ + 4j>, = o, CA"^ ax y = ^;c*tf-»^ 4- -^(2.r — ;c2 + .^3 4. ^cV^logx) 10. xii - A-)g + (I - A-)£ -irxy = o, ^2^4^6'\4 6 3 5/ J > = (^ + ^log.)^(x + £. + ^^ + ^^,-^ +■■■) -^(3.-8.)+.5.[5(i-l.-i). + -i!:5l/2 _ 1 _ I + 2 _ 1 _ lU + . . I ^ 2.4.6.8V3 26^548/ J § XV.] EXAMPLES. 197 12. x^—^ -j- y = jvt, y = (A + B\ogx)(i -"^ + -^ - -4^ + . . .) \ 1.2 I.2^3 I.2^3^4 / L1.2V1 2/ i.2^3\i 23/ J V 1-3 1-3^5 1.3^5^7 / 8 _i — X 2 25 105 \ 2 , , 2.4 I — — x-^ -\ -i — X- 3.9 34-9-II / 14. Express the particular integral of the equation (a) in the form of an ascending series ; (/?) in the form of a descending series ; (y) as a finite expression. [See Example XIV. 9, for the complementary function.] W y = -(-^^log. + 5- - ^^ + f .- - '-!.-' 5 V 6 6.7 y 198 THE HYPERGEOMETRIC SERIES. [Art. 1 8 1. CHAPTER VIII. THfe HYPERGEOMETRIC SERIES. XVI. \ General Solution of the Binomial Equation of the Second Order, 181. The symbol i^(a, /?, y, z) is used to denote the series ^ ^ gjg^ ^ a(a4.l)^(^+l) ^^ ^ a(a-|- 1) (a + 2)^(^+ 1) (/?4-2) ^, ^ ^^^ i.y l.2.y(y+i) i.2.3.y(y+i) (y+2) which is known as the hypergeometric series. Regarding the first three elements, a, yS, and y, as constants, and the fourth as a variable containing x, the series includes a great variety of functions of x. In fact we shall now show that one, and generally both, of the independent integrals of a binomial differential equation of the second order whose second member is zero can be expressed by means of hypergeometric series in which the variable element is a power of x. 182. Using the notation of Art. 123, X— = ^, whence x"— = ^(^ - i), dx dx" ^ ' we may, as in Art. 167 (first multiplying by a suitable power of x)y reduce the binomial equation to the form /(^)7 4- x^^{{^)y = 0, (i) §XVI.] BINOMIAL EQUATION OF THE SECOND ORDER. 1 99 in which / and <^ are algebraic functions, one of which will be of a degree corresponding to the order of the equation, and the other of the same or an inferior degree. If the equation is of the second order, it may be written (^ _ a) (O - d)y - ^x'{h- — c) {& - d)y ^ o, . . (2) in which q and s are positive or negative constants. Further- more, the equation is readily reduced to a form in which q and s are each equal to unity ; for, putting d we have z = qx^, and ^' = _ dz ^' = ^x' ^ = i^, or & = sd-' qsx^ - ^dx s and, substituting, equation (2) becomes {*'-7){^'-7)--<^--:)(^-7>=°-- -(3) 183. We may, therefore, suppose the binomial equation of the second order reduced to the standard form (t9- - a){d- - b)y - x{d- - c){& - d)y = o. . . (i) Substituting in this equation y = %'^ArX^ + ^, we have ^"^Arl^m + r—a) {m-\-r—d)x^+'>'—{m-\-r—c) (rn-\-r—d)x»f+*'+^'] =0, and, equating to zero the coefficient of x^ + % {m-\-r—a)(m-\-r—b)Ar—{m-{-r—i—c){fn-\-r—i—d)Ar-i = o. 200 THE HYPERGEOMETRIC SERIES. [Art. 1 83. This gives the relation between consecutive coefficients, and, when r =z o, (m — a){m — b)Ao = o ; whence m = a or m =z b. Putting m =.a,vft have for the first integral >' = -"(' + V^iVf ^ \ i{a - b -{- 1) ^ i.2{a-b->ti){a-b+2) ^ + ---j» W and, interchanging a and b, the second integral is \ i{d - a + 1) ^^ - c){b - c + i){b - d){b - d + 1) \ .. ■^ i,2{b-a+i){b-a + 2) T.-.y. u; Thus, putting a ^ c = a 1 ^ - ^ = ^ , (4) « — <5 + I = 7 J the first integral is \ i.y 1.2.7(7 +1) / = :^i^(a,A7,^), (5) and the second may be written y, = a^Fip!, ft', Y,x), (6) § XVI.] DIFFERENTIAL EQUATION OF THE SERIES. 201 where a = b — c =a-|-i— y P' = b-d = /3 -f- I _ y [ , . . . . . (7) y'=ib — a-\-i= 2 — 7 and b = a -{• 1 — y. Differential Equation of the Hypergeometric Series. 184. If in equation (i) of the preceding article we put ^ = o, and introduce a, y3, and y in place of b^ c^ and d by means of equations (4), we obtain ^(^ - I +7)j-:v(^ + a)(^ + y8);; = o, . . . (i) or, since % = x—- and &^ = x^— [- ;r-— , in the ordinary no- dx dx^ dx tation A:(i-^)gH-[y-^(i+a + «]£-ay8);=o/. . (2) This is, therefore, the differential equation of the hypergeometric series, Fia, y8, y, x). Putting, also, ^ = o in the value of y^^ we have ;; = AF{a,l^,y,x) + Bx'-yF{a -f i - y, ^ + i _ y, 2 - y, ^) for the complete integral of equation (2). Since the complete integral of the standard form of the bi- nomial equation of the second order, (i) Art. 183, is the product of this complete integral by x"^, it follows that the general binomial equation of the second order, equation (2), Art. 182, is reducible to the equation of the hypergeometric series in v and 2 by the transformations ^ = ^x^ and j/ = z^v. 202 THE HYPERGEOMETRTC SERIES. [Art. 1 85. Integral Values of y and y\ 185. When a ■=■ b \n equation (i), Art. 183, y = y'= i, and the integrals y^ and y^ become identical, so that there is but one integral in the form of a hypergeometric series. Again, if a and b differ by an integer, one of the series fails by reason of the occurrence of infinite coefficients. In this case, let a denote the greater of the two quantities, then y is an integer greater than unity, and y' is zero or a negative integer. The coefificient of x*'-^^ in F{a!y ^, y\ x), is (g + i-y) . ♦ . (g + ^-i-y) (/3+1-7) • - (jg + ^-i-y) ^ {n-i)\{2-y){s-y) ...{n-y) This is the coefficient oi x^ + ''-^, that is, oi x^ + '^-y iny^, and is the first which becomes infinite when y = n. Now, putting and denoting the sum of the preceding terms of jj (which do not become infinite when k = o) by T, the complete integral may be written in which — is the product of B^ and the coefficient written above, /I so that, when /i = o, B has the finite value B = B (a-hi-n)...(a-i){ft+i-n)...(^-i) . . {n-i)\{2-n)is-n)..,(-i) "^ ^ Putting § XVI.] INTEGRAL VALUES OF y AND y'. 203 we have, as in Arts. 172 and 175, j^j = f(o)j and, expanding in powers of h, equation (i) becomes y = Aoy, + BoT+ |(i +h\ogx + ...)[;;, + hxl;' {p) +...]; or, putting A for A^ + y and j' for ^\o), y=^Ay,+BoT+ByAogx + By'-^...,. . .(4) in which we have retained all the terms which do not vanish with /i. To find ^' or xf/\o), we have, from equation (3), ah whence, putting /^ = o, '• = -[r?(M-T- ;)-■■■]■ (5) Finally, writing the complete integral (4) in the form y = Ay^ + B-T], and taking the value of B^ from equation (3), we have, for the second integral, (a+l-7)...(a-i)(^+l-7)...(^'-l) where y^ is the first integral x'^F{a, /?, y, x), T the terms which do not become infinite in the usual expression for the second integral, and j^' the supplementary series given in equation (5). It is to be noticed that when y = i, T = o. 186. In this general solution of the case in which y is an integer, the supplementary series y' is the same as the first 204 THE HYPERGEOMETRIC SERIES. [Art. 1 86. integral j„ except that each coefficient is multiplied by a quantity which may be called its adjunct. The adjunct consists of the sum of the reciprocals of the factors in the numerator diminished by the like sum for the factors in the denominator. The first term in^^ must be regarded as having the adjunct zero. If y^ is a finite series, it is to be noticed that the adjunct of each of the vanishing terms is infinite and equal to the reciprocal of the vanishing factor. Thus the corresponding terms of the supplementary series do not vanish, but are precisely as written in the expression for ^„ except that the zero factors in the numerators are omitted. 187. As an illustration, let us take the equation ^""^ " ^'^S+ ^^"^^ 2 " ^' ~ ^^^-^ " °' which, when written in the form (i). Art. 183, is (^ - i);' - oc{^' - 9)y = o, so that ^ = I, ^ = — I, ^ = 3, <^ = — 3 ; whence a = —2, )8 = 4, y = 3. We have, therefore, \ 1-3 1.2.3.4 / 2. — 1.0.A.C.6 ./ , 1.7 , \ / \ 1.2.3^4.5 V 4.6 / + -^- in which the terms following the first three vanish. For the other integral, employing equation (6), Art. 185, because y is an integer, we have V = yi^ogx x-'(i + —^x] + y', -4.-3.2.3 \ -I.I / § XVI.] IMAGINARY VALUES OF a AND /?. 205 where the next term in the expression for T would be infinite. The part of y' corresponding to the actual terms of j, is L 1.3 \ 2 4 3/ 1.2.3 \ 2 5 2 3/ J and the part corresponding to the vanishing terms in equation (i) is as therein written, with the zero omitted. Thus we have 3 3 and • 17 =^y,\ogx - — +/, 36^ 9 where y' = i§^3_47^3 + lA^ + ii7^+ LM:8^3 + . , 1. 9 9 3 L 4.6 4.5-6.7 J Imaginary Values of a and /?. 188. We have assumed the roots a and b of f{&) =0 to be real, but the roots c and d of ({>) ~ o may be imaginary. In that case a and y3 will be conjugate imaginary quantities, say a = fjL -\- t'v, 13 = fx — tv. The integrals will then take the form L 1.7 1-2.7(7 +1) J and y^ = ^^-vfi + (M + I - 7); + ^ ^ L 1(2-7) + [(/^ + I - 7)^ + v^ir(M + 2 - y)' -h v^1 ^^, _^ 1 1.2(2 - y)(3 - y) •••J* 206 THE HYPERGEOMETRIC SERIES. [Art. 1 88. Again, when y is an integer, making the same substitutions in equation (6), Art. 185, the second integral becomes where Infinite Values of a and yS. 189. As explained in Art. 165, the function f{%) must be of the second degree, but - e) (& -f)y = o. One of its three independent integrals is •^' '^K^ ^.h.''^ 1.2.8(8 + i)c(. + I) "+■■■}' where a^ a ^ dj fi = a — e, y — a — /, 8 = « — <5 4- I, c = <2 — Y" + I, and the other two are the result of interchanging a and ^, and a and c respectively.* The notation /^f*^'/^' ^' xA has been employed for the series involved in the value of y^ above. * When two of the roots a, <5, and c of /(i9) differ by an integer, so that one of the quantities d or £ is an integer, the powers of x which occur in one of the three integrals will occur in another with infinite coefficients. By the process employed in Art. 185 these infinite terms are replaced by terms involving \ogx and the adjuncts. If both 6 and t are integers, the third integral contains terms which occur in each of the others, with doubly infinite coefficients, and by a similar process these may be replaced by terms involving (log;f)2as well as \o%x. Similar results hold for binomial equations of any order. See American Journal of Mathematics^ vol. xi., PP- 49» S0» 51' 210 THE HYPERGEOMETRIC SERIES. [Art. 1 94. Development of the Solution in Descending Series, 194. When both of the functions / and <^ in the binomial equation are of the second degree, that is, when a and y8 are finite, the integrals y^ and y^ are convergent for values of x less than unity, and divergent when x is greater than unity. In the latter case, convergent series are obtained by developing in descending powers of Xj or what is the same thing, ascending powers of x-^. Putting, in equation (i). Art. 183, 2 = -, whence ^ = 2— = —&, X dz we have (^ -h ^)(^ + d)y - z{^ +a){d' + b)y = o; hence the results are obtained by changing ^, by c, and d, in the preceding results, to —Cy —d, —a, and —b. Making these changes in equations (4), and denoting the new values of a, ft and y by a,, /?„ and y,, we find tti =5 —c •\- a = a, A = ~^H-^ = a+ I - y, 7i=— ^ + ^+l=a+l--^; and the integrals are P; = z-cF{a,, ft, y„ 2) = x'fU, a + I - y, a + I - ft iV . . . . (i) y. = z-dF{p.U ft', y/, 2) = x^fU, ^ + I - y, ^ + I _ a, i^ (2) § XVL] TRANSFORMATION OF THE EQUATION. 211 Transformation of the Equation of the Hypergeometric Series. 195. The equation of the hypergeometric series, :.(i-:^)g+[7-^(i+a+/3)]g-a^;; = o, .(i) admits of transformation in a variety of ways into equations of the same form, leading to other integrals still expressed by means of hypergeometric series. One such transformation is obviously y ■=. x^-'*v\ for, since this will give an equation for v one of whose integrals is the simple hypergeometric series F(a', ^, y', x), that is, the trans- formed equation is of the form (i), the new values of a, ^8, and y being a' = a H- I — y, )8' = )8 + I - r, 7' = 2 - y. The second integral of this equation will be v^ = x^-yF{a' + I - r', /?' + I - 7', 2 - 7', x) = xy--F{a, ^, 7, x), and the corresponding value of y is F{a, /3, 7, x), which is the value of y^. This transformation, therefore, gives no new integral. 196. Let us now make a transformation of the form ^ =a (i ~ x)f^ V. 212 THE HYPERGEOMETRIC SERIES. [Art. 1 96. Comparing equation (i) with the form we have %^'t^<^y = - x{\ — X) X \ — X and e = ^(i — a:) Hence, putting in the formulae of Art. 152 «;, = (i — xY^ we find, for the transformed equation, „ _ 7 j_ 7-g-^-i-2/^ ^^ ■" ^ "^ \ - X ^* (l - a:)^ I - .^ ^ -_ /^(/* - i) - /^(7 - <^ - ^ - i) _ HlL±-2^, (i — xY x{i — ^)* In order that (2i n^ay take the same form as 2, let ^ be so determined that the first term of this expression vanishes. This gives /a = o (in which case no transformation is effected), or else II — y — a — p. Then, if a„ /3i, 7, have the same relation to P^ and (2i that a, ft 7 have to P and (2> the form of P^ shows that y^ = 7, and tti + ^i = g + )3 + 2/A, and that of Qt shows that ttxft = a/3 + /X7. § XVI.] TRANSFORMATION OF THE EQUATION 213 Substituting the value of /x above, and solving, we find a, = 7 — a, ^x = 7 - A yi = 7. The integrals of the transformed equation are, therefore,. v^ = Fici - a, 7 - A 7. X), and V^ = X-'-'lFiT. — a, 1 — p, 2 — y, x)j V2 being derived from v^ by the same changes which convert y^ into J2. Denoting the corresponding values of y by y^ and y^y we have thus four integrals involving hypergeometric series of which the variable element is x ; namely, y, = F{a, ft y, x)y y^ = x^-yF{a + I - y, y8 + I - y, 2 - y, a:), y,= (1 - x)y — PF(y - a, y - ft y, ^), y^ = ^i-v(i — ^)Y-«-Pir(i _ a, I — ft 2 — y, ^). 197. The integral y^ is the product of two series involving powers of x with positive integral exponents, each of which has unity for its first term, and is convergent when x < i. It follows that j/3 is a series of the same form. But, from the process of integration in series, we know that there can be but one integral of this form, namely ji. Hence we have the theorem, j^i = y^, or F(a, ft y, ^) = (I - x)y — PF(y - a, y - ft y, x). In like manner y2 = J4. 214 ^^^^ HYPERGEOMETRIC SERIES. [Art. I98. Change of the Independent Variable. 198. It is obvious from the form of the equation of the hypergeometric series, equation (i), Art. 195, that, if we change the independent variable from x to / = I — a:, the first and third terms will be unchanged, and the second will be unchanged in form. The result is /(I - /)^ + [i 4- a -f ^ - 7 - /(I + a + /?)] ^ -^ a^;; = o ; comparing this with the original equation, we find that a and ^ are unchanged, while y is replaced by I + a + ^8 - 7. We hence derive the integral jFs = 7^(a, /?, 1+ a + y8 - 7, I - X), A comparison of this integral with y^ or F{a., y3, 7, x) shows that from any integral expressed by a hypergeometric series we can derive another integral of the same equation by making the above-mentioned change in the third element, and at the same time changing the fourth or variable element to what may be called its complement^ that is, the result of subtracting it from unity. The process applies equally well to an integral of the form y = wv^ where z; is a hypergeometric series ; for a new integral of the equation for v gives a new integral of the equation for J. Thus the integrals 72* jJ^s. and j^ would lead to the three new integrals y^y ye, and ys, which will be found in a subsequent article. § XVI.] CHANGE OF THE INDEPENDENT VARIABLE. 21 5 199. It is shown in Art. 194 that, when we change the independent variable to ^ = -, the binomial equation retains its form. The equation considered in that article becomes that of the hypergeometric series when we put ^ = o, whence c = —a. Thus equation (i), Art. 194, gives the integral Y^ = X-^'Ffa, a + I — y, a + I — ^, - j A comparison of this integral with F{a, /?, y, x) gives a method by which we may pass from any integral in the form of a hypergeometric series to another in which the variable element is replaced by its reciprocal ; and, as in the preceding article, the process applies also to an integral of the form J/ z= wv, where ^' is a hypergeometric series. 200. If we start with the variable x, and alternately take the complement and the reciprocal, we obtain the following six values of the variable, X i-x ' ^ I -^ I I — X 1 — X X X the seventh term of the series being identical with the first. The corresponding integrals derived from 7, by the processes of Arts. 198 and 199, are y^ = ^(s A y, X ), ys = ^(«» A I + a + )8 - 7, 1-x) , = (I - x)-<^F(^a, y - A r, -73^) > X-'^fL, a+i-r, I+a + i8-y, -^-^) » ^-«7^^a, a+l~y, a+I-ft ^ \, y^3 ^21 = 2l6 THE HYPERGEOMETRIC SERIES. [Art. 200. where, in writing the last two, the constant factor (— i)-* has been omitted. 201. From each of the integrals given above we may derive three others, exactly as j^z, /s and y^ are derived from y^. We have thus the following system of twenty-four integrals, y^ = F{o., ^, y, X), y^ = x^-yF{a + I - y, ^ + I - y, 2 - y, ^), ^3 = (i-x)y — ^F{y-a, y- A y, x), y^ = x'-y{i-x)y---PF{i-a, I -A 2-y, x), y^ = F(a, ft, I + a + ^ - y, i-x)y ye = (i - x)y---PF{y - a, y - ^ i _ a - /? + y, i - x), y, = x—yF{a + I - y, /3 + I - y, I + a + ^ - y, i - ^), ^/g = ^^-Y(l -x)y-'^-^F{l-a, I -13, I -a-/3-fy, I - ^), j;, = (I - x)—fU y - a a + I - a ^3^^ , j-.o = (I - x)-^f(b, y _ a, /? + I - a, -^^ , y^^ = {-xy-y{i-x)y-^—F(i-(3, a+i-y, a+i-/J, YZri)' y,, = (_:v)l-7(I_^)v-x-^7^/I_a, )8-fi-y, /3+i-a, ^^] , ,, = ^i-y(l _^)v-i-«ir/a+l-y, 1-^2-7, -73^)' ^^,5 = (I - ^)-Pi^(y - a, A 7, -73^) > § XVI.] THE TWENTY-FOUR INTEGRALS. 21/ jVx7 = x-'^FU., a+l-y, i+a + y8-y, - ^ ~ "^ j , ;;,9 = ^-^i^/y8 + I - y, A i+a+/?-y, -^^) , = ^-«/^^a, a + I — y, a + I — A -J , = x-^fU, ^ + I _ y, ^ + I ^ a, iV j3^ = ^«-v(^ _ i)Y-«-P7^/i - a, y - a, y8 + I - a, -^ Since the binomial equation of the second order can be transformed into the equation of the hypergeometric series, it follows that the binomial equation has in general twenty-four integrals expressible by means of hypergeometric series.* But, in the cases considered in Arts. 189 and 190, where a or /? is infinite, we have only the integrals y^ and y^. * The twenty-four integrals are written above exactly as they arise in the process indicated, except that the factor ( — \Y~'^ is dropped in the case of J14 and /i6, and (_i)Y — *-^ is dropped in the case of jj/is and jao- Because y-i = js and y^ ■=. y^, the first and third integral of each group are equal, and so also are the second and fourth, the omission of a factor in the cases mentioned above causing no exception. It may also be shown, by comparing the developments in powers of x, that the integrals of the first group are respectively equal to those of the fourth group, and those of the second to those of the fifth group. But in the third and sixth groups jg = {— i)";y2i and jio = ( — 1)^^22- Thus the twenty-four integrals consist of six sets of equal quantities, as follows : — 2l8 THE HYPERGEOMETRIC SERIES. [Art. 202. Solutions in Finite Form. 202. The condition that jF(a, /?, y, x) may represent a finite series is readily seen to be that one of the elements a or /8 shall be zero or a negative integer. But, since y^ = 73, the form of J3 shows that, if either y — a or y — y3 is zero * or a negative integer, /^(a, y8, y, ;r) may be expressed in finite algebraic form. For example, one integral of the equation 2x{\--x)^-\-{\ — {2n-\- <,)x\-J-- iny ^ o is the infinite series represented by i^(|, «, J, x). Here y -- a is a negative integer, and, using the form y^ the integral may be written (i _ x)-^--F{-i, i - n, i, x), yi = yi = yi3 = yis, j^2 = y4 = yu = yi6j ys = y? = yn = yi^^ ye = yt = yis = J20, ^10 = J12 = (-i)^>'22 = i-ifyu- Between any three integrals belonging to different sets there must exist a relation of the form y^ = Afy^ + JVjy^. These relations, in which the values of M and iV involve Gamma Functions, are equivalent to those given by Gauss in the memoir " Determinatio Seriei Nostrae per Aequationem Differentialem Secundi Ordinis," IVerke, vol. iii. See equations [86], p. 213, and [93], p. 220. The twenty-four integrals, and their separation into six sets of equal quantities, were first given by Kummer, in a memoir " Ueber die hypergeometrische Reihe," Crelle^ vol. XV., p. 52. The order of the integrals is different from that given above, and some errors involving factors of the form (—1)'^ occur in the statement of the equalities. The values of M and N are given by Kummer for the integrals numbered by him i, 3, 5, 7, 13, and 14, corresponding to the integrals 71, j2> yi^ y6f ^9. and >'io above. * The case in which y — = o has already been considered in Art. 191. § XVL] SOLUTIONS IN FINITE FORM. 219 in which the second factor is the finite series I H ^'^—^ -X = I + \2n — i)^. Hence the integral in question is (i — x)» + ^ 203. In like manner the integral ^2 will be a finite series if either of the quantities a+i—yoryS+i — yis zero or a negative integer ; and, since ^2 = J4> the form of j/^ shows that if either i — a or i — /? is zero or a negative integer (in other words, if a or y8 is a positive integer), j/2 may be expressed in finite form. It will be noticed that the eight quantities, «; /3, y — a, y — /3, a + I — 7, ^ + I — y, I — a, 1 — jS, are the only values of the first two elements in the twenty-four integrals ; hence the only cases in which they furnish finite in- tegrals are those in which either a, /?, y — a, or y — ^ is an integer. In the case of the general binomial equation of the second order, the condition given in the preceding article, when applied to both integrals, is sufficient to determine whether finite algebraic solutions exist.* * Finite solutions involving transcendental functions occur in certain cases considered in the following chapter. See Arts. 209, 213, 214, and 217. 220 THE HYPERGEOMETRIC SERIES. [Art. 203. Examples XVI. 1. Show that, in the notation of the hypergeometric series, (/ + uy + (/ - uy = 2tnF[-\n, -\n + J, \, ^^ , (/ + uy - {t - uy = 2ntn-^uFl-\n + J, - 1« + i, f, ^^ , \og{i -^ x) = xF{iy I, 2, -a:), logi-^ = 2XF{\, I, f, A-), f-= i^^i, >^, I, 1^ = I + xfI^i, k, 2, |^ = I + a: + ^A:*i^( I, ^, 3, -J = etc., where ^ = 00, sin^ = xF\k, k\ f, —\ , k = k' = c^, cos:v = 7^^^, /&; i, - -^^ , k = U=:z^, cosh^ = j^^^, k', i, -^V k=.k'=^, sin-^AT = xF{\, J, f, ^2), tan-^:x: = xF(^\, I, f, -x^), 2. Show that £7^(a,ft y, ^) = ^ir(a + i, ^ + i, y + i, ^), £F(a,P,y,x) = ^(l±^M^i.(a + 2,/3 + 2,y + 2,.),etc. OV* y(y + i) § XVL] EXAMPLES. 221 3. Show that the equation Ay Ji^iB -^ Cx)^ -\-{D +Ex + Fx')p^ = o ax do^ can be reduced to the equation of the hypergeometric series, and hence that the complete integral is A where a and b are the roots oi D -\- Ex -f- Fx^ = o, a)3 = — , F a + ;8 + I = f , y = f + ^^^ and y' = ;|-^^ , the two 7^ (^ — b)F {b — d)F independent integrals being related as y^ is to y^ in Art. 198. 4. Find the particular integral of the equation (^ - a){^ - b)y - x{& - c){& - d)y = kxP, and derive the integrals in Art. 183 from the result. Solve the following equations : — 5. x(i - x)^ + (I - 2^)g -~iy = o, y = AF{h, h i, X) + Bx-\ = -^^'"-)g + ^ . \x 6. 2J£:(l — x)-^ + x-/ — y = o, dx^ dx y^x{A-\-B\ogx)-\-Bl2 + ^-x^ + ^"^ + h^J^ 4- . . . V \ 4 4-62 4.6.8 3 / 232 THE HYPERGEOMETRIC SERIES. [Art. 203. 7. Transform the series , 8 , 8.10 - , 8. 10.12 , , ^ = I + 2-x + 3 ^ + 4 ^^ + . . . 9 9.11 9-II-I3 by means of the theorem of Art. 197. Solve, in finite form, the following equations : — -)-5#,+(- = 0, y^ ^ I + 6* + *» + B (I- -xY -)S+*('- - 2^)| + ¥^ = -- 0, 9. ;r(i ax- y = A(i-' x)^(i - 6x) H- Bx^S - 6a:). 10. 2:c(i - a:)^ 4. ^ + 4^ = o, ^::c* ax y =x ^(3 - i2a: + 8a^) + ^A;i(i - x)^, 11. Solve the equation first transforming to the new independent variable z = 1 — ^. y = A{i - x^)i + ^^(i - x^)^. 12. When a is a negative integer, the six integrals of Art. 200 are all finite series, and therefore must, in that case, be all multiples of the integrally,. Verify this when a = —i. 13. Show, by comparing the first two terms of the development, that 7, = ^,3, and thence that F{a, ft y, sin* 6) = (cos* e)y---^F{y - a, y - ft 7, sin* 6) = (sec* 6YF{a, y - ft y, -tan* 6) = {sQc^eyFly - a, ft y, -tan*^). § XVI.] EXAMPLES. 223 14. From the expression for sin-'^ as a hypergeometric series, derive ^ = sin^/^(iif, sin^^) = sin^cos^i^(i, I, I, sin*^) = \xciBF{\, i,f, -tan'^). 15. The integrals of the equation are sin nB and cos nB ; form the equation in which ^ = sin ^ is th6 independent variable, and thence derive four expressions, as in Ex. 13, for each of these quantities. sin«^ = n%mBFi^\ - \n,\-\- \n, f, sin*^), = ;2sin^cos^i^(i ■\- \n^ \ — \n, f, sin^^), = «sin^(cos^)«-^i^(i -\n,\- \n, |, -tan*^), = /2sin^(cos^)-«-^i^(i + |«, J + \n, f, -tan*^); cos«^ = F{—\n, ^n, ^, sin*^), = cos BFii + ^n,^ - in, J, sin* 6), . . =. (cos ByFi-^n, J - i«, i, -tan* B), = (cos B)—F(in, i + ^n, |, -tan*0). 1 6. Denoting by i? the expression ^(^ - i)^ + (3^ - i)f + x^ dx^ ax show that the equation xt— + (i-\- xx-f\R = o is equivalent to dx \ dxj ^(^ - i)^ + 3^(3^ - i)^ + (19^ - i)^ + 8^'« = o, ax^)e^^^^ + c{i + 3«jc*)^-3«^' We may, if we please, express this solution in a logarithmic form ; for, solving for Cy we have ^ T ■ — ^.» , 230 RICCATPS EQUATION. [Art. 209. whence lax^y + x^y + 3^* 210. In like manner, if q is the reciprocal of a negative odd integer, u^ and //6 give finite independent integrals. Hence we have a complete solution in finite terms, when q is of the form —: , where k is any integer. Substituting this expression in w = 2^ — 2, we have 2 — 4^ 2^+1 2/^+1 where ^ is any integer. Changing the sign of k, this expression becomes —r^ — ; so that the condition of integrability in finite form for equation (2), Art. 204, is that m should be of the form 2k ± i' where k is zero or a positive integer. Relations between the Six Integrals. 211. Since ?^3, Art. 208, is an integral of equation (i), it must be of the form Au^ -\- Bu^, where Uj and U2 are the inte- grals given in Art. 206. Now ^ q ^ ^« 2 ! ^ ^3 3 ! and «3 is the product of this series and the series a g , a^ -ig — 1 x^^ q q^ 2q — 1 2 \ § XVII.] RELATIONS BETWEEN THE SIX INTEGRALS. 23 1 This product is a series having for its first term unity, and proceeding by integral powers of x^. But u^ is a series involving these powers, while in general tt^ contains none of these powers. It follows that, putting ti^ = Ati^^ + Bti^, we must have B = o and A = I ; that is, u^ = ti^, the odd powers of x^ vanishing in the product. In like manner we can show that u^ = ti^, and that ?/6 = 2^4 = t^2' 212. It thus appears that u^ and u^ are not independent integrals, but merely different expressions for the same func- tion ; nevertheless we have, in Art. 209, derived from them independent integrals in the case where they furnish finite expressions, namely, when ^ is the reciprocal of a positive odd integer. The explanation is that the finite expressions are nof the actual values of 71^ and n^, in these cases, but the results of rejecting from the series involved the infinite series of terms in which the vanishing factor occurs in the denominator as well as in the numerator of each coefficient. The rejected part of the series Vj_, Art. 208, is a multiple of the series v^ ; so that the finite expression, which we may denote by [u^], differs from ii^ by a multiple of u^. The expansion of the complete product u^ is still the series //i, consisting of even powers only of x^ ; but that of the product [^3] contains also odd powers of x^. These odd powers are accompanied by odd powers of a ; hence, since [u^] is the result of changing the sign of a in [u^], it is evident that we shall have For example, when ^ = i, [u^] = ^•^, and [u^] = ^-'^•^, of which the expansions contain both even and odd powers of ;ir, but Uj, is the even function cosher = ^(e*^^ + ^-''^). In like manner, when g is the reciprocal of a negative odd integer, we have the independent integrals [u^] and [ue], and «2 = iM+iM. 232 TRANSFORMATIONS OF [Art. 2 1 3. Transformations of RiccaiiU Equation. 213. Certain important differential equations may be derived by transforming Riccati's equation, — — a^x'^^-^u = o (i) dx-^ For this purpose it is convenient to use the t?-form of the equa- tion, namely, {}{n — i)u — a^x'^^u = o (2) Let us first change the independent variable from x to ^, where z = mx^^ whence t>' = 2— = -»9. dz q The result is tn'^ putting m = -J and writing »? and x in place of ^?' and Zy this becomes ^'^{f^ — m)u — a^x^u=^ o, (3) which in the ordinary notation is d^u _ in — I ^ _ 2 _ / \ dx"^ X dx I X Hence, putting ^ = — , and writing — in place of x^ in the in nt six values of ti given in Arts. 206 and 208, we have the follow- ing six integrals of equation (4), a^ x^ , a* X* m — 2 2 {m — 2) {m — 4) 2^2 ! \ m -\t 2 2 (w 4- 2) (/« 4- 4) 2*2 ! J § XVIL] RICCATPS EQUATION. 233 «3 = e'^^i I ax + ) (^^ ^ — - - . . . , \ m — 1 {m—i){m — 2) 2\ J 2^. = e^^x*" [ I ' — ax + -^ ■ — f-^ !— ^ — ... I, * \ m + i {m + 1) {m -\- 2) 2! y ^■r-f , m — 1 , (m — i){m — -i) a^x^ , \ \ m—i {m—i){m — 2)2\ J U(, = e-'^^x^i I H ! — ax + -'^ ' — '— -^^ 1- . . . ). \ m -\- 1 {m + 1) {m -^ 2) 2 \ J The factor m^ has been omitted in writing u^, u^, and ue, but we still have ti^, = u^ = u^, and ti^ = u^ — U(,. Equation (4) is integrable in finite terms when tn is an odd integer, the complete integral being A\ti^ ■{■ B\u^ when m is. positive, and A [?/ J + B\ti^ when m is negative. 214. If in equation (3) v/e put m z=z 2p -\- i, and make the transformation u = x^v^ we have, since y'^x^ V=px^ V+x^f^ V= x^{n +/>)V, ('"^ +/) ('^ — / — 0^ — ^''^^^ = 05 which in the ordinary notation is ^2^ p(p 4. i) dx^ x^ ' This equation is integrable in finite terms when / is an integer.* The case in which p=2 occurs in investigations concerning the figure of the earth. * See the memoir " On RiccATi's Equation and its Transformations, and on some Definite Integrals which satisfy them," by J. W. L. Glaisher, Philosophical Transac- tions for 1881, in which the six integrals of this equation are dediiced directly, and those of the equations treated in the preceding articles are derived from them. 234 BESSEVS EQUATION. [Art. 2 1 5. BesseVs Equation, 215. If, in equation (3), Art. 213, we put w = 2« and ^' = — I, and make the transformation u —x*^y, the result is (,9» _ «.)^ + ^2j, = o, (i) or, in the ordinary notation, which is known as BesseVs Equation. Making the substitutions in the values of u^ and /^j, Art. 213, and denoting the corre- sponding integrals of Bessel's equation by^^ and _;/_„, we have « / ^ x'^ , I ^4 \ •^'' V « + I 2» («+ l)(« + 2) 24.2 ! " */ y-n = x-^[\ + -+ — -+ .... \ « — I 2' («— l)(«— 2) 24.2! J It will be noticed that either of these integrals may be obtained from the other by changing the sign of n, which we are at liberty to do by virtue of the form of the differential equation. 216. The integrals corresponding to the other four values of u in Art. 213 are imaginary in form. Making the substitutions in the value of w^, we may write, since //^ = ti^. = ^O'w* yn — x» (cos X -\-i sin x) {Pn — iQn) » in which P - T _ (2^4- i)(2;z + 3) x^ . (2« + l)(2« -h 2) 2 ! Q^^ 2n-\-i ^ {2n-^i){2n + 7,){2n + ^) x^ ^ 2«H-I (2« + l)(2« + 2)(2« 4- 3) 3 ! § XVI I.] FINITE SOLUTIONS. 2$$ The value of f„ derived from ue is the same thing with the sign of i changed ; hence we infer that y„ = x» {Fn cos X + Q„ sin x) , and also that Pn sin X — Qn cos X = o* Changing the sign of n, the other integral of Bessel's equa- tion may, in Hke manner, be written in the form y-n= X-^'iP-n cos X+ Q-nSiux), where {2n — i) yzn — 2) 2 ! ^ _ 2n-\ {2n- i){2n-z){2n-s) x^ 2« — I (2« — l)(2« — 2)(2« — 3) 3 ! i7«/V'_„, when we use the finite expressions, as in Art. 217, commences with the term containing x'^*^. Denoting the coefficient of this term by A, the rejected part oiy^^t is Ay„. Thus y-„= [ J_ „] + Ay^ = 77, + ^■7?2 + Ay„. A But we have shown that tjj =>'_„; hence 7/3 = — -y^, where A is the coefficient of x^*^ in P—n — iQ—ny ^^^* ^^* ^^ "~ ^Q—n' "^^^ ^^^> since 2w is an odd integer. Thus (2»-l)!(2«)! '* § XVII.] THE BESSELIAN FUNCTION. 23/ The Besselian Function. 219. If, when ;^ is a positive integer, we multiply j„, Art. 215, by the constant — L_, the resulting integral of Bessel's 2«/2 ! equation is known as the Besselian function of the n\h order, and is denoted byy«. Thus T x^ f I x^ , I :r4 2"n \\ « + I 22 (^n -\- i) {n + 2) 242 ! = 2°° (- O"* /^f Y "*" ^*' ° {n + r)\r\\2j More generally, for all values of n we may write / — ^" / _ I ^ J I x^ __ \ ~" 2«r(« H- l) \ « + I 22 (« + l) (« + 2) 242 ! ' ' 'J and then, in general, the complete integral of Bessel's equation is where y_„ is the same function of — n thaty„ is of n. It is to be noticed, however, that the factor which converts the series j/_„ to y_„ is zero in value when « is a positive integer. Substituting the values of rj^ and }f„, Arts. 217 and 215, we have, for the devel- opment of the odd function smxlP_„'] — cos;r[^_„], (2« — l)! (2»)! \ 2(2« + 2) 2.4(2« + 2)(2« + 4) / 238 BESSEVS EQUATION. [Art. 2 1 9. The series in this case contains infinite terms which are thus rendered finite, while the finite terms preceding that which contains x^*" are made to vanish. The result is that, when n is an integer, /_„ =(-!)«/«, and the expression AJ„ + BJ-n fails to represent the complete integral. The second integral in this case takes the logarithmic form, and is found in Art. 221. 220. The expression forj^„, given in Art. 216, shows that where ■^- = ?axfTT)(^"'°" + ^" ''"">'• 2« -+- 2 2 ! (2« + 2) (2« + 4) 4 ! Q^-x ^^ + 5 -^3 Y (2^ + 7) (2^ + 9) £l_ 2« + 2 3 ! (2«+ 2)(2« + 4) 5 ! ♦ Finite expressions fory„ and /_„ exist when « is of the form / 4- ^, / being an integer. These are multiples of tJj and tJj, Art. 217, respectively. Substituting in the numerical factors the values of the corresponding Gamma functions, which are (2/>+i)! r(/+ !) = ip^\') r(/+ D = Jp+^p^ ^^> and, taking account also, in the case of A + ^, of the factor found in the preceding foot-note, we find /.^, _ C2/>)! sin^[/>_^^^j^)]-cos.r[^_(^^^)] '^■'^"2/-i/!V^ ^>*+i and § XVIL] BESSELIAN FUNCTION OF THE SECOND KIND. 239 The Besselian Function of the Second Kind, 221. The second integral, when n is an integer, may be found by the process employed in Arts. 175 and 176, and in similar cases. Thus, changing equation (i), Art. 215, to {^ — n){d--\-n — h)y + x^y = o, and putting ^ = 2^^^;r'« + *'', the relation between consecutive coefficients is Ar=- (w + 2r — «) {m -^ 2r ■\- n ■— A') where h^ is put for h. Making m = n and m^—n^h suc- cessively, we have the integrals ^„_^-. . ' .0 in-\- 2 -K) 2.4(2^ + 2- -/^' ){z„ + ^-h') " and y-n^x- •n + A J T^ + h- X^ .h^)(^2n-2- ■h) + ... 4_ x-'^ ' (^ H-/^- K). ..(2« + h-h^){2n — 2 -h)...{. -h) .(, x^ {2n -\- 2 -{• h — ■h')(2+h) + ...' Denoting the product of x*" and the series last written by ^{h)^ we have j/'(o) = jk«, and the complete integral y = A^yn -f- Boy^*,;, may be written y = Aoyn + ^0^+ f (I + /^ log^ + ...)[>'« + '^'A'(o) +...], 240 BESSEVS EQUATION. [Art. 221. where, when h = o, B^ - 2.4 .. . 2n{2n — 2){27i — 4) ... 2 2^*^~^n\ {n — \)\ and T denotes the aggregate of terms in ^_„, which remain finite when ^ = o. We have therefore y^Ayn^- BoT+ Byn \ogx + B^\,\o) + . . . , and may take as the second integral, when ^ = o, yn\o%x— 2««-^«!(« — i)!7'+«A'(o)- If this expression be divided by 2«« !, the first term becomes y„log;ir; denoting the quotient by F„, and developing j/''(o) as in Art. 173, we have K=/JogA:-2«-»(«-i)!:r-«ri+-^^ L « — I 2« , I JC4_ I £«»2l__"l («~i)(« — 2) 24.2! "* (;? — i) ! 2»'»-»(« — i)!j 2«+»«!Li.(« + 1)\ n-\-i)2^ I.2.(« + l)(« + 2) V 2 ;«+I «4-2/24 J and the complete integral of Bessel's equation, when n is an integer may be written y = AJn + BK^ § > VII.] LEGENDRE'S EQUATION. 24 1 The integral F„ is called the Besselian function of the second kind.-^ Legendre's Equation. 22J. The equation {1 — x^)-^ — 2x^ + n{n + i)y = o^ . . . . (i) ax^ ax or, as it may be written, is known as Legendres EquatioUy because, when n is an integer, it is the differential equation satisfied by the nXh member of a set of rational integral functions of x known as the Legendrean Coefficients.! Particular interest, therefore, attaches to the case in which « is a positive integer; and it is to be noticed * The properties of the Besselian functions are discussed in Lommel's " Studien iiber die Bessel'schen Functionen," Leipzig, 1868; Todhunter's "Treatise on La- place's Functions, Lame's Functions, and Bessel's Functions," London, 1875, etc. t The Legendrean Coefficient of the wth order is the coefficient of a« in the expansion in ascending powers of a of the expression y/{l — 2ax + a2) and is denoted by Pn{x), or simply by P„. It is readity shown that whence, substituting F= ^.^anPn and equating to z.ero the coefficient of tt», we find "When X =1, V= — i— = 1 + o + o* + . . . ; hence Pn{i) = i for all values of n. 242 LEGENDRES EQUATION. [Art. 222. that this includes the case in which « is a negative integer; for, if in that case we put — n =s n' + i, whence — (« -f i) = «', we shall have an equation of the same form in which n' is zero or a positive integer. 223. When written in the t9-form, Legendre's equation is t?(t^ - i)^ - a:='(t9 - «) (i> + « 4- 1)^ = o, a binomial equation in which both terms are of the second degree in 1% Hence the equation may be solved in series pro- ceeding either by ascending or descending powers of x. Putting jy = S^-^r^*"^"** we have, for the integrals in ascending series, ^'x = I - /«(« -f i)^+ «(« - 2)(« + i){n + 3)^- . . ., 2! 4! and ^,=a:-(«-i)(;^ + 2)^'+(«~i)(«-3)(« + ^)(«+4)^-.... Again, writing the equation in the form and putting y = ^^ArX*"'^'', we have the integrals in descend- ing series \ 2(2«— l) 2.4(2«-- l)(2« — 3) / and y, = X-n-^ (l + (^+l)(^ + 2) ^., \ 2(2;? + 3) ^ (n + i)(n -h 2)(n -^ 3)(n -\- 4) ^^_^ ^ ^ ^ \ 2.4(2« + 3)(2«-|-5) '*'/ § XVIL] THE LEGENDREAN COEFFICIENTS. 243 The Legendrean Coefficients, 224. When « is a positive integer, j^, or y^ is a finite series according as n is even or odd ; and in either case J3 is a finite expression, differing ixomy^ ox y^ only by a constant factor. If y^ be multiplied by the constant {2n- i){2n'- 3) ... I Qj. {2n)\ n\ 2«(«!)"' the resulting integral is the Legendrean coefficient of the «th order, which is denoted by P„. By the cancellation of common factors in the numerators and denominators of the coefficients, the successive values of Pn may be written as follows : — 2 ^ 4.2 4.2 4.2' ^ 4.2 4.2 4.2 ' 11^ 6 2:1:5 4 J. 1:5:^ , 5-3-i ^6 - 6.4.2 ^ 3 6.4.2^ "^ 3 6.4.2^ ~ 6.4.2' in which the law of formation of the coefficients is obvious.* * The constant is so taken that the definition of P„ given above agrees with that given in the preceding foot-note. For, putting x = i, and forming the differ- ences of the successive fractions which in the expressions last written are multipHed by the binomial coefficients, it is readily shown that P„{i)=i,iot all values of n. 244 I.EGENDRES EQUATION. [Art. 225. The Second Integral when n is an Integer. 225. When n is an integer, the second integral of Legendre's equation admits of expression in a finite form. Assume y=zuP„-v, (i) where u and v are functions of x. By substitution in equation (i), Art. 222, we have « j (I - x-)"^ - 2^^ j^n{n-^i)pA + 2{i- x-)^ ^ K ax ax ) ax ax -\-Pn\ (1-^=')- 2x— ^-(i-jc»)- — \-2x—-n{n-\-\)v^o, (. dx"^ dx ) dx^ dx in which the coefficient of u vanishes, because P„ is an integral, and that of P^ will vanish if u be so taken that f ^^d^u du (l — XA- 2X—- = O. ^ ' dx^ dx du This condition is satisfied if we take (i — ;t:*) — - = i, whence dx " = *1°Sj3T' • • • W the equation then becomes and we shall have a solution of Legendre's equation in the assumed form (i), if v is determined as a particular integral of this equation. § XVI I.] THE SECOND INTEGRAL. 245 Now, since /*„ is a rational and integral algebraic function of the «th degree, the second member of equation (3) is an algebraic function of the (« — i)th degree ; hence the particular integral required is the sum of those of several equations of the form (i — x"^)-^ — 2x-^ + n{n + i)y = axP^ . . . (4) dx^ ax in which / is a positive integer less than n. Solving equation (4) in descending series, the particular integral is ~ {J>-n){J> + n-\-i)\ {p-^n-i){p-n-2)^ + /(/-i)(/-2)(/-3) ^-4 + .. V {p-\.n-i){p-\-n-z){p-n-2){p-n-^) / which, when / is an integer, is a finite series containing no negative powers of x. Thus the particular integral of equation (4) is an algebraic function of x of the pth. degree, and that of equation (3) is an algebraic function of the {n — i)th degree. Denoting this function by i?„, we have therefore an integral of Legendre's equation of the form 226. Since (2« = iAiog^-ie« (5) X — 1 X 3^3 gjcs the product ^Pn log , when developed in descending series, X """ 1 commences with the term containing x*^~^', and as Rn contains no terms of higher degree, the development of Q„ cannot con- tain x^. It follows that, putting Q„ = Ay^ -h By^ where y^ and y^ are the integrals in descending series. Art. 223, we must 246 LEGENDRE'S EQUATION. [Art. 2261 have Qn = By^* But j4 commences with the term x-'^-^y we therefore infer that in the product above mentioned the terms with positive exponents are the same as those of /?„, and are cancelled thereby in the development of Qn, while the terms with negative exponents vanish until we reach the term Bx-*^^^. The formation of the required terms of this product affords a ready method of calculating R„.^( * To determine the value of B, we notice that equation (3), Art. 147, gives, for the relation between the integrals Pn and Qn of Legendre's equation, /.„^-^„^ = _^l_, (r) dx dx I ^ xi where y4 is a definite constant. Substituting from equation (5), this gives P„^ + (;r» - 1) \Pn~^ ^Rn~^'\= A. L dx dx J Putting x—i, we have A=i, because P«(i)= i, and Pn and Pn being rational integral functions, the quantity in brackets does not become infinite. 'Now, from Art. 224, Pn = ^^^^' J3 ; substituting this value, and putting A = i, Qn = By^^, equation (i) becomes B(^2n) \ l dy^ ^ dy-A _ i 2«(« ! )2 V ^ dx ^^dx ) ~ i^ x^' Developing both members in descending powers, and comparing the first terms, we whence z?_ 2*'(n\y . B: C2W+I) t The Legendrean coefficients are sometimes called zonal harmonics, the term spherical harmonics (in French and German treatises /onciions spheriques and Kugel- functionen) being applied to a more general class of functions which include them. The function Qn is the zonal harmonic of the second kind. Discussions of the properties of the functions Pn and Qn will be found in Todhunter's Treatise " On Laplace's Functions, Lame's Functions, and Bessel's Functions," London, 1875 ; Ferrers' "Spherical Harmonics," London, 1877; Heine's "Handbuch der Kugel- functionen," Berlin, 1878 ; etc. § XVII.] EXAMPLES. 247 Examples XVII. Solve the following differential equations : — dy , a^ . (x — d)e^^~^-{- c(x •{■ a^e-*^^'^ 1. -^4.j;2 = — , s y=- ^^ ~ •• dx x^ x'^{e(^^~'^ 4- ce-<^^~'^) d^u _ 8 2. -; a^x ^u = o, dx^ u = Axe-^''''~\i + 3«;c-*) +^^^3«^~^(i - s^^'^)- d'^u 2 du 3. ~ ~-^a^u = o, u = Ae^^(i—ax) +Be-'*-^(i -i-ax). dx^ X dx dx"^ X dx ^ d^u , 2 du , ^ -, cos (x — a) d^u 4 du 2 — dx^ X dx ^ u =:Ax-^e^'^(i — ax) -\- Bx-^g-^^(i -{- ax). d^u , A du , * dx^^ X dx ' u =z Ax-^{cosax + axsinax) +Bx-^{sinax — axQOsax). ^^2 -^ x^ y = Ax~^e"^{i — ax -i-ia^x^) +Bx-^e-^^{i -\- ax + ^a^x^). d^y , 6v y = C^-2[(3 — «2^2) COS («;ic + a) + 3;2;t: sin {fix + a)]. ^48 RICCATrS EQUATION, ETC. [Art. 226. v = C(;r "COS h^ *sin 1. ^ \ a a ] y = Cj>;~^[(i — ^^») cos (^ + a) + ^ sin {x + a)]. 13. Show that, when q is the reciprocal of an odd integer, the integral of Riccati's equation, a^A:*?-*^ = o, dx'' may be written in the form 14. Show that for all values of n , , n -\- 2 x^ (n -Y 2){n -\- a;) x-i ■^ "^w + i 2!"^ (;z+i)(« + 2) 3!"^ ••• ^ajTa- Ljf 1-^4. ^ + ^ ^ _ (^ + 2)(;?4-4) ^ , ■^«+ I 2! («+ l)(«+2) 3!"^ *•• 15. Show that the complete integral of the equation dx^ dx x-^ may be written in the form xy = A{2 — qx) ■\' Be-9^{2-\- qx). § XVII.] EXAMPLES. 249 16. If in Riccati's equation a^ = — i, show that the integral may be expressed in Besselian functions. «=V.[^A(l.) + ^/_.(i^)] 17. Reduce to Bessel's form the equation x-^'tl 4. nx% + (^ + cx'''^)y - o, dx^ ■ dx and show that its integral in Besselian functions is 2m d^y 20. •^^+^+;' = 0» ;' = 4/o(2^i)+^Ko(2^i). ^2y ,{u,v) = {a,d) = C is a relation between x, y, z and the arbitrary constant C, and is therefore an integral. This is, in fact, the general expression for the integrals of the system of which u — a and v = b are two independent integrals. Accordingly, it will be found that, if u and v are functions of x and y satisfying equation (2) of the preceding article, <^ (w, v) also satisfies that equation, <^ being an arbitrary function. 256 SIMULTANEOUS EQUATIONS [Art. 233. Equations of Higher Order equivalent to Determinate Systems of the First Order, 233. An equation of the second order may be regarded as equivalent to two equations of the first order between Xj y and /, one of which is that which defines /, namely, and the other is the result of writing ^ in place of — ^ in the dx dx^ given equation. For example, the system equivalent to the equation which is solved in Art. 'j^y is, when written in the symmetrical form of Art. 227, '-l^dx^-% p y in which the equation involving ^ and ^ is independent of x, and thus directly integrable. The integrals of the equivalent system are the same as the first integrals of the equation of the second order, of which two, corresponding to the constants of integration employed, may be regarded as independent. Compare Art. 79. The complete integral of the equation of the second order, containing as *it does both constants of integration, is an integral equation, but not an integral, being the result of eliminating the variable / either before or after a second integration. Compare Art. 82. In like manner, an equation of the «th order is equivalent to a system of n equations of the first order, between n -h i varia- bles. Again, two simultaneous equations of the second order § XVIIL] GEOMETRICAL INTERPRETATION. 25/ between three variables are equivalent to a system of four equations of the first order between five variables, and so on. Geometrical Meaning of a System involving Three Variables. 234. Let X, y and z be regarded as the rectangular coor- dinates in space of a moving point ; then, since the system of differential equations dx _dy _dz X ~Y~Z determines the ratios of dx, dy and dz, it determines at every instant the direction in which the point (;r, y, z), subject to the differential equations, is moving. Starting, then, from any initial point A, the moving point will describe a definite line, and any two equations between x, y and z, representing two surfaces of which this line is the intersection, will form a parti- cular solution. If we take a point not on the line thus deter- mined for a new initial point, we shall determine another line in space representing another particular solution. The two equa- tions forming the complete solution must contain two arbitrary constants, so that it may be possible to give any initial position to {x, y, z). The entire system of lines representing particular solutions is therefore a doubly infinite system of lines, no two of which can intersect, assuming X, Y and Z to be one-valued functions, because at each position there is but one direction in which the point {x, y, z) can move. We hence infer also that the constants will appear only in the first degree. 235. Consider, now, the complete solution as given by two integral equations between x, y, z and the constants a and d. The surfaces represented determine by their intersection a par- ticular line of the system. Let the constant ^ pass through all possible values, while a remains fixed ; then at least one of the surfaces moves, and the intersection describes a surface. The 2S8 SIMULTANEOUS EQUATIONS. [Art. 235. equation of this surface is the integral corresponding to the con- stant a ; for it is the result of eliminating b from the two equa- tions, and is thus a relation between x, y, z and a. Hence, an integral represents a surface passing through a singly infinite system of lines selected from the doubly, infinite system, and of course not intersecting any of the other lines of the system.* If a and b both vary but in such a manner that C = (a, b) remains constant, the intersection of the two surfaces describes the surface whose equation is the integral corresponding to the constant C. Compare Art. 232. 236. Thus, in the example given in Art. 230, the integral (4) represents a plane perpendicular to the line I m n and the integral (5) represents a sphere whose centre is at the origin. The intersection of the plane and sphere corresponding to particular values of the constants is a circle having its centre upon, and its plane perpendicular to, the fixed line (i). Hence the doubly infinite system of lines represented by the differential equations (i). Art. 230, consists of the circles which have this line for axis ; and the integrals of the differential system represent all surfaces of revolution having the same line for axis. Examples XVHI. Solve the following systems of simultaneous equations : — I. ^=^==_^, y^ + ^^a, \ogbx^i^n-^L X z y z * On the other hand, of the surface represented by an integral equation, we can only say that it passes through a particular line of the system. § XVIIL] EXAMPLES. 259 ' dx 2X _ dy , , 2X dx dy X — - -\ — X -\- y = be^. y -\- z z-\- X X -\- y dx _ dy _ dz X^ — y2 — 2* 2Xy 2XZ Idx _ mdy _ slix^y^z)^ — y X — z ndz y = az, X* -\-y^ + z^ = bz. i^x H- m^y + n^z = aj mn{y — z) nl{z^x) lm{x — yY I'^x'' -^ ni^y^ ■\- n^z=. b. r adx _ bdy _ cdz ax^ + by^ 4- ^z'^ = A, {b — c)yz (c — a)zx {a — b)xy a'^x'^ + b'^y^ -\- c^^ = B, ^ dx _dy _ dz X y z — asjix"^ -\- y^ •\- z"^^ y — ax, x^^'' =^ ^\z-\-sj{p(^ -\-y^-\-^)\ 8. Show that the general integral of dx _dy _dz I m n represents cylindrical surfaces, and that the general integral of dx _ dy _ dz x-a~ y- P^ z^y represents conical surfaces. 26o SIMULTANEOUS EQUATIONS [Art. 237. XIX. Simultaneous Linear Equations. 237. We have seen that the complete solution of a system of simultaneous equations of the first order between n -\- \ variables consists of « relations between the n -\-\ variables and n constants of integration. Selecting any two variables, the elimination of the remaining n — \ variables gives a rela- tion between these two variables, involving in general the n constants. We may also, selecting one of the two variables as inde- pendent, perform the elimination before the integration, the result being the equation of the «th order,* of which the equa- tion just mentioned is the complete integral. For example, in the case of three variables, x^ y and t, if we require the differential equation connecting x with the inde- pendent variable /, the two given equations are to be regarded as connecting with t the four quantities x, y, — and -^. Taking their derivatives with respect to t, we have four equa- dx dv d^x d^v tions containing x, r, -r-, -r-, —r- and —r- ; and from these ^ -" dt dt dt' dt^ ' dv dv^ four we can eliminate y, -j- and -5-, thus obtaining an equa- at dt^ tion of the second order, in which x is the dependent, and t the independent variable. 238. As a method of solution the process is particularly applicable to linear equations with constant coefficients, since * The differential equation connecting two of the variables may be of a lower order, in which case the integral relation will contain fewer than n constants. For example, one of the equations of the first order may contain only two variables, as in Art. 228, and then the integral relation will contain but one constant. § XIX.] LINEAR SYSTEMS. 261 in that case we have a direct method of solving the resulting equations. For example, the equations and f + 5^+;^ = ^' ....... (I) f^-x + sy = e^^ (2) are linear equations with constant coefficients, if t be taken as the independent variable. Differentiating the first equation, we have and since — ^ does not occur in this it is unnecessary to differ- entiate the second. Eliminating -j- and j/ by means of equa- tions (2) and (i), we have The complementary function is (A -\- Bt)e~^*, and the par- ticular integral is found by the methods of section X. The resulting value of x is x={A + Bt) e-^ + ^et - ^e'^, and, substituting this value in equation (i), we find without further integration, y=-(A-hB+Bi) e-^i+i^e-t + i^eK 262 SIMULTANEOUS EQUATIONS. [Art. 239. 239. The differentiation and elimination required in the process illustrated above are more expeditiously performed by the symbolic method. For, since the differentiation is indi- cated by symbolic multiplication by Z>, the equations may be treated as ordinary algebraic equations. Moreover, the process is the same if one or both the equations are of an order higher than the first. For example, the system d'^y dx . dx . dy « when written symbolically, is {2D^- a,)y-Dx=2t, 2Dy -f (4Z) — 3)^ = o. Eliminating Xy we have, in the determinant notation, 2D 4^ - 3 I o 4^ — 3 or (Z)-l)»(2Z> + 3)^=2-f/, and integrating, y=^{A+Bt)e^ + Ce-i^-^t. The value of x is, in this example, most readily derived from that of y by first eliminating Dx from the given equations, thus obtaining {^D^ -h 2D- i6)y - 3^ = 8/, whence, substituting the value of j/, X = ^ — 4 2/ 2D aD-z 2D The complementary functions for the two variables will then be of the same form, and will involve two sets of constants. By substituting in one of the given equations, we shall have an identity in which, equating to zero the coefficients of the several terms of the complementary function, the relations between the constants may be determined. 241. The number of constants of integration which enter the solution is that which indicates the order of the resultant equation. This number is not necessarily the sum of the in- dices of the orders of the given equations, although it cannot exceed this sum ; it depends upon the form of the given equa- tions, being, as the process shows, the index of the degree in D of the determinant of the first members. Denoting this number by m, the values of the n dependent variables contain n sets of m constants, of which one set is arbitrary. Substituting the values in one of the given equa- tions, we have an identity giving m relations between the con- stants ; it is therefore necessary to substitute in « — i of the given equations to obtain the relations between the constants. 264 SIMUL TANEOUS EQUA TIONS. [Art. 242. Introduction of a New Variable. 242. The solution of a system of differential equations is sometimes facilitated by the introduction of a new variable, in terms of which we then seek to express each of the original variables. Given, for example, the system dx dy dz , >, where X-ax-^-by-^-cz-ird, V= a'x + b'y + c'z + d', Z=a"x-{-b"y-\-c"z-\-d"^ If we introduce a new variable t by assuming dt equal to the common value of the members of equation (i), we shall have the system dx dy dz ,. . . Z=y=z=*' ('> involving four variables, which is linear if t be taken as the independent variable. Writing the equations symbolically, the system is {a — D)x -\- by -\- ez + d =0, a'x + {b' - D)y + c'z + ^' = o, a^^x + ^'> + (c" - D)z -f- ^" = o ; (3) whence a-D b'-D D d b c d} b — D c' d' /^" ^"- D (4) § XIX.] INTRODUCTION OF A NEW VARIABLE. 265 in which D may be omitted in the second member because it contains no variable. Denoting the roots of the cubic a-D b a' b'-D (5) by Xi, Xa and A.3, equation (4) and the similar equations for y and z give y = A^e^-' J^B'e^-' +Ce^^' +k' L . . . . (6) z = A"e^-' + ^"^^^' + C"/«' H- /^" J in which ky k\ k" are the values of ;r, /, 3 respectively, which make X = o, V=o and Z = o. Substituting these values in the first of equations (3), we have one of the three equations determining k, k' and k", and for the constants of integration the three relations, {a - K)A + bA' + ^A" = o, {a-\^)B-\-bB'-{-cB"=o, (a-X^)C-^bC'-{-cC"^o. In like manner, substitution in each of the other equations gives three relations between the constants, making in all nine relations, of which six are independent. The three relations between A, A^ and ^" are (^-Xx)^ + ^^' + ^^"=o, a^A + (b'-K)A'-\-c'A" = o, a"A-\-b"A'-h{c"-X,)A"=o, 266 SIMULTANEOUS EQUATIONS. [Art. 242. which are equivalent to two equations for the ratios A :A':A'\ since their determinant vanishes because Aj is a root of equa- tion (5). 243. The introduction of a new variable, as in the preceding article, introduces a new constant of integration into the system, but this constant is so connected with the new variable that the relations between the original variables obtained by eliminating the new variable are independent also of this constant. Thus in the value of x, equation (6), we might have put / + a in place of /, employing only two other constants ; then the relations between x, y and 2, which we should obtain by eliminating /, would obviously contain only the two constants last mentioned. Examples XIX. Solve the following systems of linear equations : — at at dx dy ,. 2. = — ^— = dt^ Zx—y x-\-y x={A+ Bt)e^*, y=(A-B-\- Bt)e'^. 3* (Sy + 9^)^*^ + dy + dz = o, (4JV + 3^)^-^ -{- 2dy — dz = o, y = Ae-^+Be-7^^ z = — ^Ae-"^ ^Be-?"^, 4. ^=^/=^, -- my mx X = A cos mt '\-B sin mt, y = A sin mt — B cos mt. § XIX.] EXAMPLES. 267 ;. a —- -\' n^y = e"^ , -f- S. a- + n^y=^e', - + «^ = o, az= — nAe»^ + nBe-""^ — «*— I 6. ^ + ^ay = o, 4^ - m^x = o, dt^ ^ df^ mx tux = .^^(. y = g^ ( A^ sm A^cos y/2 y/2 mx wa: ^ . mx cos ^, sin — - N/2 7- i dx . dy . . . 4 ^ + 9 ^ + 44^ + 49;^ = ^> 3^ + 7^ + 34^+38^'=^', X = ^^-"^ + Be-^i 4- -^// - ¥ - -^^^ y= — Ae-* + 4^^- 6' — ^i-t + V + ^^= 12 — 3{x,y) = c. In fact, the condition of integrability, Art. 245, reduces in this case to dS_dT^^ dy dx which is the same as the condition of exactness for the differ- ential expression Sdx -^^ Tdy. See Art. 25. 249. The most obvious application of this principle is to the case in which one variable can be entirely separated from the other two. Thus the example in Art. 246 might have been solved in this way ; for, dividing by zy^, which separates the variable Zy it becomes y dx — xd'v dz ; = 0, § XX.] HOMOGENEOUS EQUATIONS. 275 an exact equation of which the integral is X log 2 = ^. \ y Homogeneous Equations, 250. In the case of a homogeneous equation between Xy y and z, one variable can be separated from the other two by means of a transformation of the same form as that employed in the corresponding case with two variables, Art. 20. For, putting X ■= zUy y — zv, the homogeneous equation may be written in the form z**(f>(u, v)dx + z"ij/(u, v)dy -{- z"x{^, v)dz = o ; and, substituting £^x = zdu -f- udZy dy = zdv -f vdz^ we have z<^{u, v)du -f z\\i {u, v)dv + [x(«) ^) + U(^{u, v) + v\\i{u,v)'\ dz = o. If the coefficient of dz vanishes, we have an equation between the two variables u and v. If not, the equation takes the form dz {u, v)du -f \\i{u, v)dv _ z x{u,v) -\-u{u,v) -{-z;if/(u,z>)~' ' and, in accordance with Art. 248, the second term will be an exact differential if the given equation is integrable. 251. As an example, let us take the equation (jj;2 -\.yz-\- z^)dx + (2= + zx -{• x^)dy -\- {x^ + xy +y')dz = o, . (i) 2/6 EQUATIONS CONTAINING [Art. 25 1. which will be found to satisfy the condition of integrability. Making the substitutions, and reducing, we have dz (f" -^ V -\- \)du -\- (^W^ -\- u -\- \)dv _ z {u -\- If + \) {uv -\- u -\- v) ~~ ' Knowing the second term to be an exact differential, we in- tegrate it at once with respect to //, and obtain log Z - log ; — - + C = o, The symmetry of this equation shows that 6^ is a constant and not a function of v : thus the integral of equation (i) is xy +yz -\- zx = {:{x -{-y -\- z). Equations containing more than Three Variables, 252. In order that an equation of the form Pdx 4- Qdy + Rdz + Tdt = o involving four variables may be integrable, it must obviously be integrable when any one of the four variables is made constant. Thus, regarding z, x and y successively as constants, equation (2), Art. 245, gives the three conditions of integrability, -(f-f)--(f-f)-{x). If, therefore, we determine all the particu- lar solutions consistent with equation (i), the result will, when f is regarded as arbitrary, include all the particular solutions. The equation which completes the solution will, as in the pre- ceding example, be found by integration, and will therefore contain an arbitrary constant C, to which a special value must 2 So EQUATIONS INVOLVING THREE VARIABLES, [Art. 257. be given (as well as a special form to the function/) in order to produce a given particular solution. 258. The general solution of the equation Pdx + Qdy -{■ Rdz z= o (i) may be presented in quite a different form, which is due to Monge, depending upon a special mode of assuming the equa- tion containing the arbitrary function. Let /x be an integrating factor of the equation Pdx -h Qdy = o when z is regarded as a constant, and let V= C be the corre- sponding integral, so that dF= fiFdx -{- fiQdy. Then, in the first place, the pair of equations z=^, and F= C, (2) where c and C are arbitrary constants, constitutes a class of particular solutions of (i). Now, for the general solution, let us assume ^=<^W (3) Differentiating, we have fxPdx-\-fiQdy-\-['^-'{z)\dz = o,. . . . (4) which, combined with equation (i), gives (^ - \^) - f^^y^ = o. (5) § XX.] MONGERS SOLUTION. 28 1 Hence, if F= ^{z) be taken as one of the relations between the variables, we must have, in order to satisfy equation (i), either dz = o, or else ^^ - <^\z) -i^R^o • . . (6) dz The first supposition gives z — c and V—{c), a system of solutions of the form (2) ; the second constitutes, in connection with equation (3), Moitges solution. It is to be noticed that when it is possible to determine <^ so that equation (6) is identically satisfied, the given equation is integrable, and V—<^{z) is its integral. But, in the non-inte- grable case, <^ is to be regarded as arbitrary. Monge's solution includes all solutions excepting those of the form (2). To show this, it is only necessary to notice that, with this exception, any particular solution can be expressed in the form xz=zf^(z), y=f^{z)\ and, substituting these values in the expression for V as a function of x, y and z, we have an equation of the form V={z) determining the form of <^ for the particular solution in question. The particular solution is therefore among those determined by one of the two methods of satisfying equation (5) ; and, as it is not of the form (2), it must be that determined by equations (3) and (6). The distinction between this solution and that given in Art. 257 is further explained in Art. 262 from the geometrical point of view. Geometrical Meaning of a Single Differential Equation between Three Variables. 259. Regarding x, y and z as the rectangular coordinates of a variable point, as in Art. 234, the single equation Pdx -f- Qdy^ Rdz = (i) l32 EQUATIONS INVOLVING THREE VARIABLES. [Art. 259. expresses that the point {Xy y^ d) is moving in some direction, of which the direction-cosines /, niy n^ which are proportional to dx% dy and dzy satisfy the condition Pl-\-Qm-\-Rn^Q (2) Consider also a point satisfying the simultaneous equations dx __dy _dz P~ Q~ R' (3) and therefore moving in the direction whose direction-cosines satisfy A = A = JL (4) P Q R ^^^ Suppose the moving points which satisfy equations (i) and (3) respectively to be passing through the same fixed point A ; then P, Q and R have the same values for each, and equations (2) and (4) give l\ + Mfi -\- nv = o, which is the condition expressing that the directions in question are at right angles. We have seen, in Art. 234, that equations (3) represent a system of lines, there being one line of the system passing through any given point. Hence equation (i) simply restricts a point to move in such a manner that it every- where cuts orthogonally the system of lines represented by equations (3), which we may call the auxiliary system. 260. Now, suppose in the first place that equation (i) is integrable. The integral represents a system of surfaces one of which passes through the given point A. This surface con- tains all the possible paths of the moving point which pass through A, and every line in space representing a particular solution lies in some one of the surfaces belonging to the system. § XX.] GEOMETRICAL INTERPRETATION. 283 The restriction imposed by equation (i) is in this case completely expressed by a single equation. Every member of the system of surfaces represented by the integral cuts the auxiliary system of lines orthogonally, so that equation (2), Art. 245, considered with reference to the system of lines represented by equations (3), expresses the condition that the system shall admit of a system of orthogonally cutting surfaces. 261. On the other hand, when the condition of integrability is not satisfied, the possible paths of the moving point which pass through A do not lie in any one surface, the auxiliary system of lines, in this case, not admitting of orthogonally cut- ting surfaces.* When, as in the example of Art. 256, the point subject to equation (i) is in addition restricted to a given surface, the auxiliary lines not piercing this surface orthogonally, there is in general at each point but one direction on the surface in which * The distinction between the two cases may be further elucidated thus : Select from the doubly infinite system of auxiliary lines those which pierce a given plane in any closed curve, thus forming a tubular surface of which the lines may be called the elements. Then, in the first case, points moving on the tubular surface and cutting the elements orthogonally will describe closed curves ; but, in the second case, they will describe spirals. The forces of a conservative system afford an example of the first or integrable case. For, if X, Y and Z are the components, in the directions of the axes, of a force whose direction and magnitude are functions of x, y and z, the lines of force are those whose differential equations are dx _ dy _ dz X ~ Y~ Z' The equation Xdx + Ydy + Zdz = will be satisfied by a particle moving perpendicularly to the lines of force, so that no work is done upon it by the force ; and this equation is integrable, the integral V= C being the equation of a system of /eve^ surfaces to which the lines of force are everywhere normal. 284 EQUATIONS INVOLVING THREE VARIABLES. [Art. 26 1. the point can move perpendicularly to the auxiliary lines. We thus have a singly infinite system of lines on the given surface, for the solution of the restricted problem. 262. In a general solution the assumed surface, as, for example, the cylindrical surface represented by equation (i), Art. 257, must be capable of passing through the line in space representing any particular solution ; and, the surface being thus properly determined, the line in question will be a member of the singly infinite system determined upon the surface by the additional integral equation found. The peculiarity of the general solution of Art. 258 is that the assumed surface V—<^{z) is made up of elements which are themselves particular solutions of a certain class. We still have a singly infinite system of particular solutions upon the assumed surface, namely, the elements just mentioned. But upon each surface there is in addition the unique solution determined by equation (6). The points on the line thus determined are excep- tions to the general rule, mentioned in the preceding article, that at each point there is but one direction on the surface in which a point can move perpendicularly to the auxiliary lines. The line is, in fact, the locus of the points at which the auxiliary lines pierce the surface orthogonally. Examples XX. Solve the following integrable equations : — 1. 2{y-\-z)dx ■\-{x-\-T^y-\-2z)dy-\- {x-^y)dz == o, (x+yy{y-\-z) = c. 2. (y — z)dx + 2{x-\- 3y — z)dy-' 2{x-\-2y)dz = o, {x-^2y){y^zy = c. 3. {a — z) {ydx ■{- xdy) -\- xydz = o, xy = ^{z — a). § XX.] EXAMPLES. 285 4. {y-\-afdx-\-zdy—{y-\-d)dz — o, z = {x -\- c) (^y ■\- a) . 5. {ay — bz) dx + {cz — ax) dy + {bx — cy) dz = o, {ax — cz) = C(^_>' — (^2) . 6. ^jc 4-^H- (-^ +^ + 2+ 1)^0 = o, (jt:+_>' + 2;)\x). § XXL] PARTIAL DIFFERENTIAL EQUATIONS. 287 CHAPTER XL PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. XXI. Equations involving a Single Partial Derivative, 263. An equation of the form Pdx + Qdy + Rdzz=Q (i) which satisfies the condition of integrabihty is sometimes called a total differential equation, because it gives the total differ- ential of one of the variables regarded as a function of the other two. Thus, if x and y be the independent variables, the equation gives P O dz •= — — dx —-^ dy. or, in the notation of partial derivatives, dz P . . Tx^-R' : '^'^ and dy R' ^^^ that is to say, we have each of the partial derivatives of 2 given in the form of a function of x, y and z. 288 PARTIAL DIFFERENTIAL EQUATIONS [Art. 263. An equation of the form (2) or (3), giving the value of a single partial derivative, or more generally an equation giving a relation between the several partial derivatives of a function of two or more independent variables, is called a partial differ- ential equation. 264. To solve a partial differential equation of the simple form (2), it is only necessary to treat it as an ordinary differential equation between x and z, y being regarded as constant, and an unknown function of y taking the place of the constant of integration. The process is the same as that of solving the total differential equation, see Art. 246, except that we have no means of determining the function of y^ which accordingly remains arbitrary. Thus the general solution of the equation contains an arbitrary function. Equations of the First Order and Degree. 265. Denoting the partial derivatives of ^ by/ and ^, thus i) -^ — — ^ dx^ dy ' a partial differential equation of the first order, in which z is the dependent and x and y the independent variables, is a rela- tion between/, q, x, y and z. A relation between x, y and ^ is a particular integral^ when the values which it and its derived equations determine for z, p and q in terms of x and y satisfy the given equation identically. We shall find that, as in the case of the simple class of equations considered in the preced- ing article, the most general solution or general integral con- tains an arbitrary function. 266. The equation of the first order and degree may be written in the form Fp + Qq = R, (i) § XXL] OF THE FIRST ORDER AND DEGREE. 289 where P, Q and R are functions of x, y and z. This is some- times called the Imear equation, the term linear, in this case, referring only to/ and q. L^' u^a, (2) in which ti is a function of x, y and z, and « is a constant, be an integral of equation (i). Taking derivatives with respect to X and y, we have du . du . ^ ^„ J du . dti — + — / = o, and — + —^ = 0; dx dz dy dz and substituting the values of / and q, hence derived in equa- tion (i), we obtain pf^ + ef^+^f = (3) dx dy dz Therefore, ii 7t — a is an integral of equation (i), u\s 3. function satisfying equation (3),* and conversely. But we have seen in Art. 231 that this equation is satisfied by the function ii when ?/ = « is an integral of the system of ordinary differential equations, dx dy dz , ^ — = ^^ = — (4) F Q R ^^^ Hence every integral of the system (4) is also an integral of equation (i). Now, it was shown in Art. 232, that if u = a and v = b * It follows from the definition of an integral that this equation is either an identity, or becomes such when z is eliminated from it by means of equation (2); but, since it does not contain the constant a which occurs in equation (2), the former alternative must be the correct one. 290 PARTIAL DIFFERENTIAL EQUATIONS. [Art. 266. are two independent integrals of the system (4), the equation /(«, v) = C includes all possible integrals of the system. Hence this equa- tion, in which / is an arbitrary function, is the general integral of equation (i). It is unnecessary to retain an arbitrary con- stant since/ is arbitrary; in fact, solving for «, the equation may be written in the form u = 4>{v)y which expresses the relation between x^ y and z with equal generality. Thus, to solve the linear equation (i), we find two inde- pendent integrals of the system (4) in the forms u = a, v = b, and then put u = {v)y where <^ is an arbitrary function. This is known as Lagrange s solution. 267. It is readily seen that we can derive in like manner the general integral of the linear partial differential equation containing more than two independent variables. Thus, the equation being ... (I) pf +pj; +. dx^ dx^ dx^ the auxiliary system is dx^ dx^ P.~ P. dxn dz ~ Pu ~R' ' ' ... (2) and, if «i = Cr, n^ = c^, . . ., n„ = c„ are independent integrals of this system, the general integral of equation (i) may be written /{u^, «2, ...««) = o, (3) where / is an arbitrary function. If an insufficient number of integrals of the system (2) is known, any one of them, or an equation involving an arbitrary function of two or more of the quantities u^, u^, . . ., u„ constitutes a particular mtegral of equation (i). § XXL] THE LAGRANGEAN LINES. 29 1 Geometrical Illustration of Lagrange'' s Solution. 268. The system of ordinary differential equations emp/oyed in Lagrange's process are sometimes called Lagrange s equations. In the case of two independent variables they represent a doubly infinite system of lines, which may be called the Lagrangean lines. We have seen in Art. 235 that every integral of the differential system represents a surface passing through Hues of the system, and not intersecting any of them. It follows, therefore, that the partial differential equation is satisfied by the equation of every surface that passes through lines of the system represented by Lagrange's equations dx _dy _dz ^ and the general integral is the general equation of the surfaces passing through lines of the system. Given, for example, the equation {mz — ny)p -\- {nx — lz)q — 1} ^ mx, . . . • (i) for which Lagrange's equations are dx __ dy _ dz mz — ny nx — Iz ly — mx (2) The integrals of this system were found, in Art. 230, to be Ix + my -\- nz = a, and x^ -\- y^ -\- z^ = 3 ; and, as stated in Art. 236, the lines represented being circles having a fixed line as axis, every integral of the system (2) 292 PARTIAL DIFFERENTIAL EQUATIONS. [Art. 268. - , , represents a surface of revolution having the same line as axis. Thus the general integral of equation (i), which is /X + my-hnz = <2 = , R^ dv dv dv dv dv dv dy dz dz dx dx dy It thus appears that the equation of which the general primitive contains a single arbitrary function is linear with respect to p and q. 7ri2. The values of P, Q and R above are called the Jacob- ians of u and v with respect to y and ^, z and Xy x and y respectively, and are denoted thus, P^ (i{u, v) d{y,zy d{u, v) d{z, x)' R d(u, v) d(x,y) The Jacobian vanishes when ?/ and v are not independent func- tions of the variables expressed in the denominator, thus R vanishes if either ?/ or ^ is a function of z only. Again, P, Q and R all vanish if 21 is expressible as a function of v. In this last case equation (i) is, in fact, reducible to v = c, which con- tains no arbitrary function. When P, Q and R are given, the functions 7C and v must be such that their Jacobians are proportional to P, Q and R, Now, if we put and V =■ b, we shall have du ■, . du , , du J — dx -\ dy -\ dz dx dy dz dv J , dv y . dv 7 — dx ■\ dy -\ dz = o\ dx dy dz 296 PARTIAL DIFFERENTIAL EQUATIONS. [Art. 272. whence, solving for the ratios dx-.dy.dzy^^ have dx __ dy _ dz d{u, v) d{u, v) d{u, v) d{y,z) d{z,x) d{x,y) Hence we shall have found proper values of // and v ii 21 — a and z; = ^ are integrals of dx _dy _dz 'p'"Q~ R ' We have thus another proof of Lagrange's solution of the linear equation. 273. In like manner, if there be n independent variables Xrj x^j . . ., x„y and one dependent variable 2, we can eliminate the arbitrary function /from the equation f{u^, u^ . . .Un) = o, in which «x> ^2? • • •> ^^» are n independent given functions of the variables. In the result of elimination the coefficient of the products of any two or more of the partial derivatives will vanish, and we shall have an equation linear in these deriva- tives, that is an equation of the form /'i/i + /'^A + • • • + Pnpn = R. Moreover, each of the coefficients P^, P^, . . ., P„ and R will be the Jacobians of u^y ti^, . . ., ?/„ with respect to 71 of the variables, and the simultaneous ordinary equations derived from ?^j = C-iy U2 ^ ^aj .,?/« = c„ will be dx-i dx2 dXn P^ P. Pn dz -R' where /*i, Pa, , . ., P„ and R are the same Jacobians. § XXL] EXAMPLES. 297 Examples XXI. Solve the following partial differential equations : — dz 1. y- 2x — 2z—y=Oy X •\-y -\- z^y'^^ipc), 2. psj{y^-x^)=y, z=y^m-^--^^{y). 3. lpA-mq==i, z=j + {/y — mx). A'p-\-q — nz, z = e"^l^ 6. yp -\- xq = z, z= {x -{-y)(}i(x^ —y''). 7. {y^x — 2x'^)p -}- (2y^ — x^y) q = gz{x^ —y^), ^"y \y^ -^v 8. xzp + yzq = xy, z^ = xy + ffi\ ^ 9. x^p-xyq-{-y^ = o, z = =^ + cf>{xy). 10. zp-\-yq = x, X + z =ycf>(x^ — z^). 11. xp + zq'-\-y = o, 12. {y + z)p+{z-^x)q = x+y, y II. xp + zq -\-y = o, tan-^^ = log:i: + (^(jV^ +02). -A.^-y {z-y)sl{x-^y^-z):==<^ y — X X \ xy j 14. x{y-z)p-^y{z-x)q=zz{x^y), xyz = cf>{x +y -\-z). 15- ^-^=7:^' (^+j)log2 = :i: + <^(^-f>'). 298 PARTIAL DIFFERENTIAL EQUATIONS. [Art. 273. 16. z — xp — yq^ a\j{x'^ ■\- y^ -\- z'^), 37. {y-^x)p-\-{y-x)q = z, z = v/(^» 4-r)<^[tan-^^ + ilog (^» +r)]- 18. J2/ + xyq — nxz, z — y"cfi {x^ — y''). 19. xy^p — y^^ + ^•^■3^ = o» log^ = h ^{p^y)' 20. (6" — :vOA + ("S* - ^a)/^ + ... +'(5 - ^„)/« = 6* — 0, where 6" = ^i + ^2 + ... + ^« + 2, <^l'S«(^x - ^), ^«(^» - ^), . . ., S^{xn - 2) J = o. dz , dz . .dz , xy dx dy dt t . ^xy ^,(y i\ 22. Find a common integral of the equations py = qx and /^ + ^j = 2. z = csl{x^ ^-J^'^)• 23. Show that ^3 + j;3 4- 03 — 3Jlr^'s = /^ is a surface of revolution, and find its axis. 24. If « = o and V = o are particular integrals of a linear partial differential equation, show that every other integral <^ = o satisfies the equation d{, u, v) ^^ d{x, y, z) ' 25. Determine the surfaces which cut orthogonally the system of similar ellipsoids , ,. m^ n^ \ z z J 26. Determine the surfaces of the second order which cut orthogo- nally the spheres :,^ + y' + z^ ^ 2ax. x'^ -\-y^ + z^ = 2by -f- 2cz. § XXIL] EQUATIONS NOT OF THE FIRST DEGREE. 299 XXII. The Non-Linear Equation of the First Order, 274. We have seen in Art. 270 that a partial differential equation of the first order may be derived from a given primi- tive by the elimination of two arbitrary constants. Such a primitive constitutes a complete integral of the differential equation ; but, when the resulting equation is linear, the general solution contains an arbitrary function which imparts a gen- erality infinitely transcending that produced by the presence of arbitrary constants or parameters. The surfaces represented by a complete integral constitute a doubly infinite system of surfaces of the same kind, while the more general class of sur- faces represented by the general integral is said to form a family of surfaces. Thus, in the example given in Art. 270, the complete integral (i) represents the doubly infinite system of planes parallel to a fixed line ; and the general integral (3) represents the family of cyHndrical surfaces whose elements are parallel to the same fixed line. 275. The differential equation derived from a complete prim- itive may be non-linear. For example, if, in the primitive, {x-hy -\-(^y-kY + z-^c% (i) h and k are regarded as arbitrary parameters, the resulting differential equation is 2^/' + ^' + =^S (2) which is not linear with respect to / and q. Equation (i) is therefore a complete integral of equation (2). Geometrically it represents a doubly infinite system of equal spheres having their centres in the plane of xy. It will be shown, however, in 300 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 275. the following articles, that the geometrical representation of the general integral of a non-linear equation is a family of surfaces equally general with that representing the general integral of a linear equation. But, since it has been shown in Art. 271 that a primitive containing an arbitrary function gives rise in all cases to a linear equation, it is obvious that the general integral of a non-linear differential. equation cannot be expressed by a single equation.* The System of Characteristics. 276. A partial differential equation of the first order, con- taining two independent variables, is of the form F{x,y,z,p, q) = (i) Let z = {x,y), (2) whence ^ = S' ^=t^ <3) be an integral ; then these values of ^, / and g satisfy equation (i) identically. If -r, j and £• be regarded as the coordinates of a point, equation (2) represents a surface. A set of correspond- ing values of x, j/, z, p and q determine not only a point upon the surface, but the direction of the tangent plane at that point, and are said to determine an element of the surface. If we per mit X and y to vary simultaneously in any manner, the corre- sponding consecutive elements of surface determine a linear * The surfaces of the same family are generated by the motion of a curve in .space, when arbitrary relati'jns exist between its parameters. The simplest case is that in which there are but two parameters ; the two equations of the curve can then be put in the form u = d, "— C2; and, if /(f 1, ^2) = o is the relation between the parameters, /(u, t^) = o is the general equation of the family. This case, therefore, corresponds to the linear differential equation. See Salmon's " Geometry of Three Dimensions," Dublin, 1874, PP 37^ e^ se^. § XXII.] THE SYSTEM OF CHARACTERISTICS. 30I elemeftt of surface ; that is, a line upon the surface together with the direction of the tangent plane at each point of the line. The linear element thus determined upon the surface (2) will in general depend upon the form of the function ^ ; but it will now be shown that, starting from any initial point upon the surface, there exists one linear element which is independent of the form of <^, provided only that equation (i) is satisfied, so that every integral surface which passes through the initial ele- ment must contain the entire linear element. 277. Let the partial derivatives of /^be denoted as follows : — = x — = y — = z — = F —z=o dx dy ^ dz ' dp ' dq Since z, p and q are functions of x and y, the derivatives of equation (i) with respect to x andjj/ give X + Z/-fPg + (2j = o, (4) Y^Zq^P^± + Q^ =0 (5) dy dy Now let X and y vary simultaneously in such a way that then, because fot every point moving in the surface dz = pdx + qdy, we have also dz ^^ pP^qQ (7) 302 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 277. Equations (6) and (7) give dx _dy^ _ dz The values of / and q in these equations being given in terms of X and J, by equations (3), they form a differential system for the variables x, y and z. Starting from any initial point {xo,yo, ^o), this system determines a line in space ; and, supposing the initial point to be taken on the surface (2), this line lies upon that surface. Now, substituting from equation (6), and remembering that dq __ d^'z _ dp dx dxdy dy^ equation (4) becomes dx dt dy dt whence g=-X-Z/ (8) In like manner, equation (5) gives f^ = -Y-Zi (9) Equations (6), (7), (8) and (9) now give dx ^dy ^ dz ^ dp ^ dq .. P Q pP+qQ X + Zp Y-^Zq' ' ^ ^ a complete differential system for the five variables x, y^ Zy p and q. Starting from any initial element of surface (^o>^o> 'S'oj/w ^o), this system determines a linear element of § XXIL] THE SYSTEM OF CHARACTERISTICS. 303 surface, and supposing the initial element to be taken on the surface (2), the entire linear element lies upon that surface. Now the system (10) is independent of the form of the func- tion , and the only restriction upon the initial element is that it must satisfy equation (i) ; it follows that every integral sur- face which contains the initial element contains the entire linear element. This linear element, depending only upon the form of equation (i), is called a characteristic of the partial differential equation. Through every element which satisfies equation (i) there passes a characteristic* 278. A complete solution of the system (10) consists of four integrals in the form of relations between x, y, z, p and q. Mul- tiplying the terms of the several fractions by X, Y, Z, —P and — Q, respectively, we obtain the exact equation dF=o, of which F= C is the integral. But it is obvious that, in order to confine our attention to the characteristics of the given equation, we must take C=o. Thus the original equation is to be taken as one of the integrals of the characteristic system. The other three integrals introduce three arbitrary constants. Hence the characteristics form a triply infinite system. For example, in the case of the equation given in Art. 275, which may be written F = p--\-q--'-+i=o, (i) 2 C"^ X = o, Y—Oy Z— — , P=2pj Q = 2qy and the equations of the characteristic are * In like manner, when there are n independent variables, a set of values of •^i, X3, . . ., xn, z, Px, P2, . . ., /«, v^^hich satisfies the differential equation, is called an element of its integral, and the consecutive series of elements determined as above are said to form a characteristic. See Jordan's "Cours d'Analyse," Paris, 1887, vol iii., pp. 318 ^^ seq. 304 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 2/8. ^ — ^ — ^g __ z^dp __ z^dq , . p^ q ~ p^ + q^~ c^p ~ c^q ^^^ Of this system, equation (i) is an integral ; the relation be- tween dp and dq gives a second integral which may be written in the form ^=/tana (3) The values of/ and q derived from equations (i) and (3) are p = co.J^^L^^, (4) Z ^=sinaV^ ^, (5) z and these equations may be taken as two of the integrals, in place of equations (i) and (3). Substituting these values in the relations between dx and dy, dx and dz respectively, we obtain, for the other two integrals, y — X tana + a;, (6) and {x%tza-\- by = c^ — z"^ (7) These last equations determine, for given values of a, a and b, the characteristic considered merely as a line, and then equa- tions (4) and (5) determine at each point the direction of the element, that is to say, the direction of a plane tangent to every integral surface which passes through the characteristic. The General Integral. 279. It follows from Art. 2^^ that every integral surface contains a singly infinite system of characteristics, so that if we make the initial element of a characteristic describe an § XXI I.] THE GENERAL INTEGRAL. 305 arbitrary line upon the surface (the Hnear element of surface along the line determining at each point the values of /o and q^, the locus of the variable characteristic will be the integral sur- face. Moreover, if we take an arbitrary line in space for the path of the initial point, it is possible so to determine p^ and q^ at each point that the characteristic shall generate an integral surface. For this purpose, we must have in the first place, ^{xo,yo,Zo,po,qo) = o (i) Again, since the path of the initial point is to lie in the surface, so that dzo = podxo + qodyo, taking the differential equations of the arbitrary curve to be dxo _ dyo _ dZo .^ ~L~M~'N' ^'^ we must have N = PoL-\-qoM, (3) where Z, M and N are functions of Xo, jTo and Zq. Geometrically, this last equation expresses the condition that the initial ele- ment must be so taken that the plane tangent to the surface shall contain the line tangent to the arbitrary curve. The general integral may now be defined as representing the family of surfaces generated by a variable characteristic having its motion thus directed by an arbitrary curve.* * That the surface thus generated is necessarily an integral will be seen in the following articles to result from the existence of a complete integral. The analytical proof requires that it be shown that, for a point moving in the surface, we have always dz = pdx + qdy, where p and q are given by the equations of the characteristic. If the common value of each member of the equations (2) be denoted by dr, the variation of t moves 306 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 279. In the case of the linear equation, when the characteristics become the Lagrangean lines, the values of po and q^ are still those which satisfy equations (i) and (3) ; but they need not be considered, because there is but one Lagrangean line through each point. Derivation of a Complete Integral from the Equations of the Characteristic. 280. The four integrals of the characteristic system contain X, y, z, /, qy and three constants. We may therefore obtain, by elimination if necessary, a relation between x,y, 2 and two of the constants. Every such equation represents, for any fixed values of the constants, a surface passing through a singly infinite sys- tem of characteristics, but not in general a system of the kind considered in Art. 279, so that the equation is not in general an integral of the partial differential equation. It will now be the characteristic, and that of t [^dt being, as iti Art. 277, the common value of each member of equations (10)] moves a point along the characteristic. The motion of a point along the surface then depends upon the two independent variables / and t. Then, since dz='^-^dt^^-^dr, dx=^-^dt^^dr, dy=^dt-\-^dr, dt dr dt dr -^ di dr and the equations of the characteristic give it remains only to prove that or that dt ^ dt ^ dt dz .dx . dy dz .dx dy tt ^ dr dr dr Letting /= o correspond to the initial point, the condition dzo = podxo + qodyo shows that the corresponding value of U is zero, tjvat is Uq = o. Consider now the value of — . This IS dt ^=^_^ ^_^^_^^_ ^ dt dtdr dt dr dtdr dt dr dtdr § XXIL] DETERMINATION OF A COMPLETE INTEGRAL. 307 shown how we may find such an integral, that is to say, since two arbitrary constants occur, a complete integral of the given equation. Suppose one integral of the characteristic system, in addition to the original equation F= o, to have been found. Let a denote the constant of integration introduced, and consider the values of p and q in terms of Xy y, z and a determined by these equa- tions. Now, in a complete solution of the characteristic system, each characteristic is particularized by a special value for each of the three constants of integration. We may distinguish those in which a has the special value a^, as the aj-characteristics ; these constitute a doubly infinite system of linear elements of surface, which together include all the point elements deter- mined by the above-mentioned values of / and q, when the par- ticular value ttj is assigned to a. Now these aj-characteristics lie upon a system of integral surfaces. To show this, consider a transverse plane of refer- But dH ^ d dz^dp dx d^x dqdy dy . dtdr dr dt dr dt ^ drdt dr dt ^ drdt ' hence » dU _dp dx .dg dy _dp dx _d^ d^^ dt ~ dr dt dr dt dt dr dt dr Substituting from the equations of the characteristic, this becomes dt dr dr dr dr dr dr or, since Zdz ■\- Xdx + Ydy -J- Pdp -f- Qdq = o, ^=-Z'^-\-pZ^ + qZ^ = ^ZCI. dt dr dr dr The integration of this gives U^ Ce-J^Zdt^ and, putting ^=0, we have C= Uq=o; hence, so long as the exponential remains finite, U= o, which was to be proved. See Jordan's " Course d' Analyse," vol. iii., p. Z^l' 308 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 280. ence. This is pierced at each point by one of the ai-characteris- tics, and at the point the element, which we may take as the initial element of the characteristic, determines in the plane of reference a direction. If, starting from any position in the plane of reference, the initial point moves in the direction thus defined, it describes a determinate curve in that plane, and the corresponding characteristic generates an integral surface. Varying the initial position in the plane of reference, we have a singly infinite system of curves in that plane, and a singly infi- nite system of integral surfaces. We have thus a system of surfaces at every point of which the values of/ and q are the values above mentioned which involve a,. Hence, if these values be substituted in the equa- tion dz =-pdx -j- qdy (which, it will be noticed, is, by Art. 277, one of the differential equations of the characteristic system), we shall have an equa- tion true at every point of this system of surfaces ; in other words, we shall have the differential equation of the system.* The integral of this equation will contain a second constant of integration /8 ; when both constants are regarded as arbitrary, it represents a doubly infinite system of surfaces containing the entire system of characteristics, and is a complete integral. 281. As an illustration, let us resume the example of Art. 278. Substitution of the values of / and q, equations (4) and (5), in dz —pdx + qdy, gives zdz J , 7 • = dx cos a -\- dy sin a. sJ{c^-2^) * It follows that the equation thus found is always integrable. This would, of course, not be generally true if the values of / and (/ simply satisfied the equation F=o. The early researches in partial differential equations were directed to the discovery of values of / and ^ which satisfied F= o and at the same time rendered dz = pdx + ^dy integrable. See Art. 294. § XXIL] DETERMINATION OF A COMPLETE INTEGRAL. 309 whence, integrating, we have 2= + (jt: cos a + 7 sin a H- /8)2 = c"^, which is therefore a complete integral of the given equation 2^/" + ^" + i) = ^^ This complete integral represents a right circular cylinder of radius ^, having its axis in the plane of xy ; and since equation (6), Art. 278, represents a plane perpendicular to the axis, we see that the characteristics in this example are equal vertical circles, with their centres in the plane of xy, regarded as elements of right cylinders. It follows that the general integral represents the family of surfaces generated by a circle of radius c, moving with its centre in, and its plane normal to, an arbitrary curve in the plane of xy. The surfaces included in the complete integral just found are those described when the arbitrary path of the centre is taken '^ a straigfht line. Relation of the General to the Complete Integral. 282. Since all the integral surfaces which pass through a given characteristic touch one another along the characteristic, and the surfaces included in a complete integral contain all the characteristics, it follows that every integral surface touches at each of its points the surface corresponding to a particular pair of values of a and y8 in the equation of the complete integral. The series of surfaces which touch a given integral surface cor- responds to a definite relation between ^ and a, say jS = <^ (a) ; thus the given integral is the envelope of the system of surfaces selected from the complete integral by putting ^={a) and so obtaining an equation containing a single arbitrary parameter. 3IO EQUATIONS NOT OF THE FIRST DEGREE. [Art. 282 The equation of the envelope of a system of surfaces repre- sented by such an equation is found in the same manner as thai of a system of curves. See Diff. Calc, Art. 365. That is to say, we ehminate the arbitrary parameter from the given equa- tion by means of its derivative with respect to this parameter. 283. For example, in the complete integral found in Art 281, if a and ^ are connected by the relation /iCOSa + ^sina + ^ = o, (i* the equation becomes 0* + [(^ — K) cos (k-\-{y — k) sin a]^ = r". ... (2) Taking the derivative with respect to a, we obtain \{x — K) cos a + (J^' — ^) sin a] [(j; — k) cos a — {x — h) sin a] = o, whence we must have either (;c — //) cos a + (>' — ^) sin a = o, .... (3) or else (j — ^) cos a — (::»: — ^) sin a = o (4) The elimination of a from equation (2) by means of equation (3) '^ives z-^c-, (5) and, in like manner, from equations (2) and (4) we obtain z^Jr{x-hy-^ {^y-kf^c- (6) Equation (i) expresses the condition that the axis of the cylin- der represented by the complete integral shall pass through the fixed point (//, k, o) ; accordingly the envelope of the system (2) consists of the planes z=±c, and the sphere (6) whose centre is § X X 1 1. ] EXPRESSION OF THE GENERAL INTE GRAL. 3 1 1 (//, k, 6). Regarding /i and k as arbitrary, equation (6) is the complete integral from which as a primitive the differential equation was derived in Art. 275. 284. To express the general integral, the relation between the constants in the complete integral must be arbitrary. Thus, the complete integral being in the form /(x,y,z, a,d)=zo, (i) we may put d = cf} (a), where <^ denotes an arbitrary function, and then the general integral is the result of eliminating a between the equations, /[x,y,z,a,^{a)^ = o, (2) and ^/[x, y, z, a, cj>(a)^ = o (3) The elimination cannot be performed until the form of <^ is spe- cified ; for, as remarked in Art. 275, the general integral cannot be expressed by a single equation unless the given partial differ- ential equation is linear. Since the general integral can thus be expressed by the aid of any complete integral, we shall hereafter regard a non-linear partial differential equation as solved when a complete integral is found. Singular Solutions. 285. There may exist a surface which at .each of its points touches one of the surfaces included in the complete inte- gral without passing through the corresponding characteristic. Every element of such a surface obviously satisfies the differen- tial equation, and its equation, not being included in the general integral, is a singiUar solution analogous to those which occur in the case of ordinary differential equations. 312 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 285. An integral surface generated, as in Art. 279, by a moving characteristic will in general touch the surface representing the singular solution along a line. If the surfaces of the complete integral have this character, the singular solution will be a part of the envelope found by the process given in the preceding- article, no matter what the form of <^ may be. In this case, equa- tions (2) and (3), Art. 284, which together determine the ultimate intersection of consecutive surfaces of the system (2), represent a characteristic and also the line of tangency with the singular solution. The former, as a varies, generates a surface belong- ing to the general integral, and the latter generates the singular solution. Thus, in the example of Art. 283, equation (3) deter- mines upon the cylinder (2) its lines of contact with the planes z^±Cy and equation (4) determines a characteristic. 286. There is, however, when a singular solution exists, a special class of integrals which touch the singular solution in single points, each of these being in fact the envelope of those members of the complete integral which pass through a given point on the singular solution. This class of integrals obviously constitutes a doubly infinite system, and thus forms a complete integral of a special kind. The complete integral (6), Art. 283, is an example. When f{x,y,z,a,b)=o is the complete integral of this special kind, the characteristics represented by equations (2) and (3), Art. 284, will, for given values of a and b, all pass through a common point, indepen- dently of the form of <^, and this point will be upon the singu- lar solution. In particular, the characteristic defined by /= o and -f.=zo will intersect that defined by y"=o and -Z = o, aa db in a point on the singular solution. Hence, in this case, the singular solution will be the result of eliminating a and b from the three equations. § XXII.] SINGULAR SOLUTIONS. 313 ^=°' i=°' i=°- It is to be noticed, however, that the eliminant of these equations may, as in the case of ordinary differential equations, include certain loci which are not solutions of the differential equation. 287. Since the characteristics which lie upon a surface of the kind considered above, all pass through the point of contact with the singular solution, it follows that the singular solution is the locus of a point such that all the characteristics which pass through it have a common element. At such a point, therefore, the initial element fails to determine the direction of the charac- teristic. Now, in the equations (10), Art. 277, the ratio dx'.dy is indeterminate only when P — o and Q = o, or when P = 00 and G = 00 ; hence one of these conditions must hold at every point of a singular solution. The former is the more usual case, so that a singular solution generally results from the elimination of/ and q from F{x,y,z,p,q) = o by means of the equations dF . dF = o and = o. d^ dq It is necessary, however, to ascertain whether the locus thus found is a solution of the differential equation, for the conditions P = o, Q = o, and P=cc, Q=oo are satisfied at certain other points besides those situated upon a singular solution ; for ex- ample, those at which all the characteristics which pass through them touch one another. In the example of Art. 278, P = o, Q = o gives the singular solution ^ = ± r, and P=oo, Q ='00 gives 2^ = o, which is the locus of the last-mentioned points, and not a solution. 314 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 288. Equations Involving p and q only, 288. We proceed to consider certain cases in which a com- plete integral is readily obtained. In the first place, let the equation be of the form ^(A^) = o (i) In this case, since X=o, F=o, Z=o, two of the equations [(10), Art. 277] of the characteristic become dp = o and dq = o\ whence p ■=i a and q ■=^ b (2) The constants a and b are not independent, for, substituting in equation (i), we have ^(^, ^) = o (3) Substituting in dz z=pdx -\- qdy, we obtain dz = adx 4" bdy ; whence, integrating, we have the complete integral z = ax + by -j- c, (4) where a and b are connected by equation (3), and r is a second arbitrary constant. 289. The characteristics in this case are straight lines, and the complete integral (4) represents a system of planes. The general integral is a developable surface. There is no singular solution. A special class of integrals which may be noticed are the envelopes of those planes belonging to the system (4) which pass through a fixed point.* These are obviously cones, whose * The characteristics which pass through a common point in all cases determine an integral surface. The integrals of this special kind constitute a triply infinite system : we may limit the common point or vertex to a fixed surface (as, for example, in Art. 286, to the singular solution), and still have a complete integral. § XXII.] EQUATIONS ANALOGOUS TO CLAIRAUT'S. 315 elements are the characteristics which pass through the fixed point. For example, if the equation is ^2 -f- ^2 = ffi'^^ these cones are right circular cones with vertical axes, and their equations are Equation Analogous to Clairaufs. 290. There is another case in which the characteristics are straight lines ; namely, when the equation is of the form z=px + qy+f{p,q) (i) In this case, X=p, V= q, Z=—i, and we have again, for two of the equations of the characteristic, dp = and dq — Q\ whence p=a, q^b (2) Substituting in dz =pdx + qdy, and integrating, we have the complete integral z = ax + by -\- c, (3) in which the constant c is not independent of a and b ; for, sub- stituting the values of p and q, equation (i) becomes z = ax-\- by -\-f{a, b), (4) which, since it is also one of the integrals of the characteristic system, must be identical with equation (3). 291. The complete integral in this case also represents a system of planes, and the general integral is a developable sur- face. A singular solution also exists. 3l6 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 29 1. For example, let the equation be z=^px-^qy^-ksj{\-\-p''-\-q^)', (i) the complete integral is z— ax^-by-^rksJ^Y-^-a^ -\-b^') (2) For the singular solution, taking the derivatives with respect to a and by we have and bk ^ ^ siix ■\- a- ■\- b-) These equations give 3 = -y ~ \j {k"" — x"" — y^y ~ v/(^2 _ ^2 _^2) and, substituting in equation (2), we have x^ + y^ -\- z"" = k^ (3) Thus the singular solution represents a sphere, the complete integral (2) its tangent planes, and the general integral the developable surface which touches the sphere along any arbi- trary curve. Equations not Containing x or y. 292. When the independent variables do not explicitly occur, the equation is of the form F{z,p,q) = o (i) § XXII.] EQUATIONS OF SPECIAL FORMS. 317 Here X—O and F=o, and the final equation of the character- istic system reduces to dp dq whence q = ap (2) Substituting in equation (i), we have F{z,p,ap) =0, the solu- tion of which gives for/ a value of the form / = (2). Thus, dz —pdx + qdy becomes dz — (f){z) (dx 4- ady) f whence, integrating, we have the complete integral, X -^ ay = \- ^ (3) The illustrative example of Arts. 278 and 281 is an instance of this form. It will be noticed that the mode of solution leads to a complete integral representing cylindrical surfaces whose elements are parallel to the plane of jry. The equation F{z, o, o) = o, representing certain planes parallel to the plane of xy, will obvi- ously be the singular solution. Equations of the Form f^{x, p)=f2{y, q). 293. When the equation does not explicitly contain z, it may be possible to separate the variables x and / from y and q, thus putting the equation in the form Mx,p)^My,q) (i) 3l8 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 293. In this case, we have Z= o, X=^ -4^, P= -4^, and the equations dx dp of the characteristic give for the relation between dx and dp^ Integrating, we have/i(;ir,/) :»^, and from equation (i), A{x,p)=My,q) = a (2) Solving these equations for/ and q, we have values of the form p = ,{x,a), ^==My,^)> and d2 =pdx + gdy becomes dz = ^(x, d)dx + <^2(7, d)dy, whence we derive the complete integral, z = Ui(^, d)dx + Ui2{y, d)dy + b. For example, let the given equation be xp-^-^-yq"^ = I. Putting xp"^ = I — yq"^ = a, we have and, integrating ds =pdx + qdy, we obtain the complete integral, z = 2sja^x + 2 sj{i — a)\Jy + b. § XXIL] CHANGE IN THE CHARACTERISTIC EQUATIONS. 319 Change of Form in the Equations of the Characteristic. 294. If we make any algebraic change in the form of the equation F{x,y,z,p, q) = o, the equations of the characteristic (10), Art. 277, will be altered. The changes, however, will be merely such modifications as might be produced by means of the equation F=o itself.* In particular, the form assumed when the equation is first solved for g may be noticed. Suppose the equation to be q = {x, y, z, p) (i) Then F=^-«^(;r,j,^,/), whence X=-^, F=-^, Z = -^, dx dy dz P — 5^, and 0=1. Putting q in the place of ^ in the partial dp derivatives, and omitting the member containing dq, the equa- tions of the characteristic become J^^dy^ ^^ = ±—, . . . . (2) dp ^ ^ dp dx^^dz a complete system for the four variables x, j/, z and /, q being the function of these variables, given by equation (i). These equations may be deduced from the consideration that the val- ues of/ and q derived from one of their integrals combined with equation (i) should render dz=pdx -\- qdy integrable.f * The complete solution of the characteristic system involving four arbitrary constants (see Art. 278) would indeed be changed, but not the special solution in which 7^= o is taken as one of the integrals. t See Boole's "Differential Equations," London, 1865, p. 336. 320 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 295. 295. As an illustration, let us take the equation z^pq, or q=- (l) Equations (2) of the preceding article become p'^dx , pdz j^ , . Z 2Z Of these the most obvious integral is p=y-\- a; whence ds =^pdx -f qdy becomes dz^{y^-d)dx + -^, from which we derive the complete integral z={y^d){x-Vb) (3) The equations of the characteristic derived from the more symmetrical form of the equation F =■ pq — z — o are dx __dy _ dz _dp _dq ^ ^ q '~ p " 2pq~ p " q' which are readily seen to be equivalent to equations (2). If the final equation of the system (4) be used, as in the process of Art. 292, to determine/ and q, we shall have /=— , q = a^Z, giving 4z==(^ + ay-\-^J, (5) another complete integral of the equation ^ =/^. § XXII.] TRANSFORMATION OF THE VARIABLES. 32 1 Transfor7nation of the Variables. 296. A partial differential equation may sometimes be re- duced by transformation of the variables to one of the forms for which complete integrals have been given in Arts. 288, 290, 292 and 293. The simplest transformation is that in which each variable is replaced by an assumed function of itself. The choice of the new variable will be suggested by the form of the given equation. Let then ^^-fi^y^-/'i^)(f/i^f/v) Hence, denoting the partial derivatives of t with respect to ^ and r] by /' and q', their expressions in terms of x,j/ and ^ are the same as if they were ordinary derivatives. For example, the equation may be written \zdxj \zdy) Putting — = d^, -^= df), ~ = dt, whence ^ = log;r, 17 = logy X y z and ^ = log ^, the equation becomes /^ -f- /^ = I (2) The complete integral of this equation is, by Art. 288, 1 = ai-\- by]-\- c, 322 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 296. •where a"^ -^ b"" =. i\ hence, putting a = cos o.y b — sin a, the com- plete integral of equation (i) is log 2 = cos a log ^ + sin a log^ + c^ or 297. In the following example the new independent variables are functions of both of the old ones. Given {x-^y-){p--\-g-) = 1 (i) Using the formulae connecting rectangular with polar coordi- nates, x — r cos B, y — r sin 0, whence we have y r^z=x'-^y^, ^ = tan-^-, dz dz /) dz sin^ p-=. — = — cos d , ^ dx dr dB r dz dz . ^ . dz cosB ^ dy dr dB r Substituting, equation (i) becomes 'dz or, putting dp = — , *-M Hence the integral is z — p cos a + ^ sin a + /8 y = \ cos a log {x"^ +JV^) + sina tan-^- + ^^ The same complete integral may be found directly by the method of characteristics (see Ex. 20). § XXIL] EXAMPLES. 323 Examples XXIL Find complete integrals for the following partial differential equations : — y i.pq=i, z = ax-\-- + b. 2. \lp -\- sjq— 2Xy ^ = i (2^ — aY + CL'^y + b. 3. /2 _ ^2 _ j^ 2 = :r sec a -\- y tana + ^. I. z = px + qy -{- J>q, z = ax -{- by + ab ; singular solution, z — — xy. S- q — xp + p^, z = axey + I o'e'^y + b. 6. y^p^ — x^q^ = x^y^, z = ^ ax"" -j- ^ {a^ — i)^;'^ + b. 7' P' + r = ^+y, z = i{x-\-ay--{-^{y-ay- + b. 8. q = ^yp"", z = ax -\- a'^y'' + b. L l I £ 9. 2 = /jj; + ^j^ -- np*^q**, z = ax-\-by~ na'^b^ ; singular solution, z — (2 — n) {xyY~\ 10. p^ - q^ = x^2Z^ z^ ^ rr -r ^)^ + (7 + ^)'' + ^. z 11. p = (qy -\- zy, yz — ax-\-2sl{ay)+b. 12. /2 _j_ ^2 _ 2^x — 2qy +1 = 0, 2Z = ;(:2 _|_jj;2 _|_ ^^(jt:2 _j_ ^) J^ y^(^y2 _ j _ ^) 13. Denoting x -\- ay by /, find a complete integral of Ex. 12 in the form 2(iH-a=)2 = /2 + /y/(/2_i_a2)_(i4.^2) iog[/+y/(/2_i-.a»)] + /J. 14. (/ + ^)(/^ + ^i') = I, v^(i +^)2= 2v'(^ ■h^jJ') +^. 324 EQUATIONS NOT OF THE FIRST DEGREE. [Art. 297. 15. x-y^z^pq- =1, ^22 = _ _i_ _ 1^ _^ ^, a^x sjy 16. x-^y^z'^p-'q = I, log— = — ^. bz 2 ay 17. p2-y2^^^y:,^^2^ z = — sin-» - H — 5lj^ i V + ^. 2 a 2 y 18. Find three complete integrals of . pq=px-^qy. 1° 22 = (^^ + a>'Y4- y8. 2° 2; = xy + 7 ^(^2 — a"^^ + <^. 19. Show directly, by comparison of the values of 2, / and q, that a surface included :^ the integral 2° can be found touching at any given point a given surface mrluded in the integral 1° ; and that the relation ^^2b-^- will then exist between the constants. Hence derive one integral from the other, as in Art. 283. Also show that the similar relations for the other pairs of integrals are y8 = 2^' 4- «'^a2, and b — d^ = aa\ 20. Show that xq — yp= a is an integral of the characteristic system for the equation {x-+y-){p- + q^) = i; and thence derive the complete integral given in Art. 297. 2 1 . Solve, by means of the transformations xy =. ^, x -\-y = v- the equation (y - ^') {qy -P^) = {P- qY- z = axy + C^i^x +7)'+ b. 22. (a:« — y^^pq — xy{p^ — q"") = i- s = ^a\og{x^ +jO H--tan-^^+<5. §XXII.] EXAMPLES. 325 23. Show that the equations of the characteristic passing through (a, )8, y) in the case of the equation Art. 289, are X — a _ y — p _Z — y a b m^ where a^ -\- b"" = m"^ -, and thence derive the special integral given in that article. 24. Deduce, in like manner, the integral formed by characteristics passing through {h, k, I) for the equation ^^ + ^^ = --1. (^ _ hy +{y- ky = lsj{c- - l^)-sj{c- - z-)Y. 25. Show that when the complete integral is of the form au + bv -^ w = o, (i) where u, v and w are rational functions of x, y and z, the elimination can be performed, giving the general integral <^ f^, ^) = o, (2) a homogeneous equation in u, v, w. Accordingly, show that the equa- tion arising from equation (i) as a primitive is the linear equation Pp -\- Qq = R, where d{y, 2) ^{y, 2) d{y, z) U 7) with similar expressions for Q and R, and that putting u^ — —', v^ =—, w lu. these values of P, Q and R agree with those derived from the general primitive in Art. 271. 326 EQUATIONS OF THE SECOND ORDER. [Aft. 298. CHAPTER XII. PARTIAL DIFFERENTIAL EQUATIONS OF HIGHER ORDER. XXIII. . Equations of the Second Order, 298. We have seen that the general solution of a partial differential equation of the first order, containing two independ- ent variables, involves an arbitrary function, although it is not possible to express the solution by a single equation except when the differential equation is linear with respect to / and q. We might thus be led to expect that the general solution of an equation of the second order could be made to depend upon two arbitrary functions. But this is not generally the case. No complete theory of the nature of a solution has yet been devel- oped, although in certain cases the general solution is expressi- ble by an equation containing two arbitrary functions. We shall consider these cases in the present section, and in the next, the important class of linear equations with constant coefficients, for which in some cases a solution of the equation of the ;zth order containing n arbitrary functions can be obtained. The Primitive containing Two Arbitrary Functions. 299. If we consider on the other hand the question of the differential equation arising from a given primitive by the elimi- nation of two arbitrary functions, we shall find that it is only in § XXIII. ] TJVO ARBITRARY FUNCTIONS. 527 certain cases that the eUmination can be performed without introducing derivatives of an order higher than the second. The general equation containing two arbitrary functions may be written in the form f{x,y,z, cfi(u), ilf(v)^ = o, in which u and v are given functions of x, y and z. The two derived equations df df ^^ = o, -^ = o, dx dy will contain 4>\ti) and ^'{v), two new unknown quantities to be eliminated. There will be three derived equations of the second order ^=0, ^^=0, ^=0, dx^ ' dxdy ' dy"" containing two new unknown quantities, <^"(?/) and «/'"(^)- We have thus in all six equations containing six unknown quantities. The elimination, therefore, cannot in general be effected.* 300. Suppose, however, that the original equation can be put in the form w = {u) +ip(v); (i) then the two derived equations of the first order, dw , dw . ,,. .(du , du \ , ,,. ^(dv , dv \ , s - + -/ = ,^ («) (^- + _/j + f (.) ^_ + -py . . (.) dw dw iu..\(du . du \ , ,,. ^(dv . dv \ , \ are independent of <^ and \\i. These, with the three derived * If we proceed to the third derivatives, we shall have ten equations and eight quantities to be eliminated, so that two equations of the third order could be found which would be satisfied by the given primitive. 328 EQUATIONS OF THE SECOND ORDER. [Art. 3CX). equations of the second order, will constitute five equations containing the four quantities \ \j/', <^", ^A"- These quantities may therefore be eliminated, the result being an equation of the second order. There is another way in which the elimination may be effected. Let one of the unknown quantities, say i/^', be elimi- nated between equations (2) and (3) ; we shall then have a single equation containing '. From this equation and its two derived equations we can eliminate <^' and <^". It is to be noticed that in this last process we meet with an intermediate equation of the first order, containing one arbitrary function. 301. Another case in which the elimination can be per- formed occurs when the primitive is of the form W= (ji{u) -^ zf\p{u), (i) in which we have two arbitrary functions of the same given function of x, y and s. In this case the derived equations take the form in which f — ), etc., are written in place of — +-^/, etc. \dx J dx dz Multiplying equations (2) and (3) by [ — jand (— j respectively, and subtracting the results, <^'(?/) and »/''(//) are eliminated to- gether, and we have again an intermediate equation of the first order containing one arbitrary function.* * The cases considered in this and the preceding article are not the only ones in which an intermediate equation of the first order can arise. See, for instance, the example given in Art. 311. § XXIIL] THE INTERMEDIATE EQUATION. 329 The Intermediate Equation of the First Order. 302. The preceding articles indicate two cases in which an intermediate equation of the first order may arise from a primi- tive. We have now to consider, on the other hand, the form of the differential equations arising from an intermediate equation of the form u = ^(v), (i) where u and v now denote given functions of x, f, Zy p and q* and <^ is an arbitrary function. Denoting the second derivatives of ^ by r, J and /, thus r = d-z dx^"* d^z dxdy the two derived equations are l^f^^ dp dq = *'(^)(l ^ dz^ dp dv 1q du du du \^^f—AJ(\ f^"^' i^'^ 4_ ^ A-—t dy dz dp dq \dy dz dp dq and the result of eliminating or, putting ^{y) in place of the function \^{y) — \y, z = ^yx'' \ogx + ^^<^(j) + ^{y). 305. Again, an equation which does not contain t may be exact * with reference to x, y being regarded as constant. Given, for example, the equation / + r + J = I j integrating, we have * The equation might also be such as to become exact with respect to the four variables p, q, z and x, by means of a factor. For this purpose three conditions of integrability would have to be satisfied; see Art. 252. This is the number of con- ditions we should expect, since by Art. 303 two must be fulfilled to render an inter- mediate integral possible, and one more is necessary to express that in that integral v= y. 332 EQUATIONS OF THE SECOND ORDER. [Art. 305. For this linear equation of the first order, Lagrange's equations are dx^dy^ ^^ , of which the first gives x-y=a, and this converts the second into dy of which the integral is evz = aey + \Xy + (y) -\- e-y\p{x —7). Mongers Method. 306. The general method of deriving an intermediate equa- tion where one exists is based upon a mode of reasoning similar to the following method for Lagrange's solution of equations of the first order, which is that by which it was originally estab- lished. Given the equation Fp + Qq = R, (i) and the differential relation dz = pdx -\- qdy, (2) § XXIIL] MONGERS METHOD. 333 which must exist when ^ is a function of x and y. Let one of the variables p and q be eUminated, thus dy or p{Pdy - Qdx) + Qdz - Rdy = o (3) Hence, the relation between x, y and z which satisfies equation (i) must be such that, when one of the two differential expres- sions occurring in equation (3) vanishes, the other will in general also vanish. Let us now write the equations Pdy - Qdx Qdz - Pdy :}• ■<" and suppose u = a, v = b,to he. two integrals of these simulta- neous equations. Then dti = o and dv = o constitute an equiva- lent differential system, and the relation between x, y and s is such that, if dti = o, then dv = o\ that is, if ti is constant, v is also constant. This condition is satisfied by putting u = <^{v), which is therefore the solution of equation (i). Geometrically the reasoning may be stated thus : If upon a surface satisfying equation (i) a point moves in such a way that Pdy — Qdx = o, then also will Qdz — Rdy = o ; that is, the point will move in one of the lines determined by equations (4). No restriction is imposed upon the surface, except that it shall pass through these lines, namely, Lagrange's lines defined by ?/ = a, V = b. The general equation of the surface so restricted is u — (v). 307. Monge applied the same reasoning to the equation Rr + Ss -h n=F, ....... (i) 334 EQUATIONS OF THE SECOND ORDER. [Art. 307. where R^ S, T and F'are functions of x, y, z, p and q, in connec- tion with which we have, for the total differentials of/ and q, dp = rdx + sdyy (2) dq = sdx + tdy (3) Eliminating two of the three variables r, s, t, we have j^dp- sdy ^^^_^ ^dq - sdx ^ y^ dx dy or Rdpdy + Tdqdx — Vdxdy = s{Rdy^ — Sdxdy + Tdx''). . . (4) If, then, we can find a relation between ;r, y, z, p and q, such that, when one of the two differential expressions contained in equation (4) vanishes, the other will vanish also, this relation will satisfy equation (i). Let us now write the equations Rdy"" — Sdydx + Tdx"" = o "| Rdpdy + Tdqdx = Vdxdy J If // = <2 and v = b are two integrals of this system, so that dti = o, and dv — o form an equivalent differential system, the required relation will be such that if du = o, then dv = o; that is, if u is constant, v is also constant. As in the preceding article this condition is fulfilled by ^ u = (v), which is now a differential equation of the first order. The integral of this equation is therefore a solution of equation (i).* * The same method applies to the more general form (3), Art. 302, when an intermediate integral exists, but the auxiliary equations are more complex. See Forsyth's Differential Equations, p. 359 ei seq. § X X 1 1 1. ] INTE GRABILIT V OF MONGERS EQUA TIONS. 335 308. The auxiliary equations (5) are known as Monges equa- tions. The first is a quadratic for the ratio dy : dx, and is there- fore decomposable into two equations of the form dy = mdx. Employing either of these the second equation becomes a rela- tion between dp, dq and dx or dy. These two equations, taken in connection with dz = pdx + qdy, form a system of three ordinary differential equations between the five variables x, y, z, p and q. Since four equations are needed to form a determinate system for five variables, it is only when a certain condition is fulfilled that it is possible to obtain by the combination of these three equations an exact equation giving an integral u = a. Again, a second condition of integrability * must be fulfilled in order that the second integral V =.b shall be possible. These two conditions are in fact the same as those mentioned in Art. 303, as necessary to the exist- ence of an intermediate integral containing an arbitrary function. 309. If R, S and T in the given equation contain x and y only, the first of Monge's equations is integrable of itself. Given, for example, the equation ocr-{x^y)s^yt^y^y{p-^q) (i) Monge's equations are xdy^ -\- {x -\-y^dydx -\-ydx'^ = Q, (2) xdpdy-\-ydqdx^^^^^^{p-q)dydx (3) * When there is a deficiency of one equation in a system, a single condition must be satisfied to make an integral possible, just as a single condition is necessary when one equation is given between three variables. Supposing one integral found, one of the variables can be completely eliminated; there is still a deficiency of one equation in the reduced system, and again a condition must be fulfilled to make a second integral possible. 336 EQUATIONS OF THE SECOND ORDER. [Art. 309. Equation (2) may be written {dy -f- dx) {xdy -\- ydx) = 0. *. Taking the second factor, we have xdy -\- ydx — o, which gives the integral xy^a, (4) and converts equation (3) into dp — dq _dx — dy p-q " x-y ' This gives for the second integral ^-^=t (5) X — y ^-^^ Hence we have for the intermediate integral ^y='i-{xy) (6) To solve this equation of the first order, Lagrange's equations are dxz=^dy= , (7) -" {x-y){xyy ^7^ of which the first gives x+y = a (8) For the second integral we readily obtain from equations (7) xdx -{- ydy = —^, {xy) whence 4>(xy)d{xy) = dz. I § XX III.] EXAMPLES OF MONGERS METHOD. 337 Since <^ is arbitrary, the integral of the first member is an arbi- trary function of xy, hence we may write z-^{xy) =(3; (9) and finally putting P = ^ (a), we have z = (xy) + il/{x-\-y), (10) which is therefore the general integral of equation (i). Another intermediate integral might have been found, but less readily, by employing the other factor of equation (2). 310. When either of the variables z, p qx q is contained in R, S ov^T, the first of Monge's equations is integrable only in connection with ds = pdz + qdy. For example, given the equation ^2^ — 2p^S -f- /2/ — o. Monge's equations are ^2^2 -f. 2pqdydx + p^dx'^ = o, and q^dpdy + p^dgdx — o. The first is a perfect square and gives only qdy + pdx = o, which converts the second into qdp — pdq = o. Hence the integrals z = a, and / = <^^> and the intermediate integral P^q4>{z). For this Lagrange's equations are 338 EQUATIONS OF THE SECOND ORDER. [Art. 3IO. , —dydz (z) o whence the integrals js = a, and X(f>(a) -i- y = p ; and, putting )8 = \l/{a), the final integral y + x. r—i= i — : (i) X +y' ^ ^ for which Monge's equations are dy^ — dx^ = 0, (2) Ap dpdy - dqdx + -^^j^dydx = o (3) Taking from equation (2) dy — dx = Oy whence the integral y=^x + a, (4) equation (3) becomes ^pdx dp— dq + -^^-1 — = o, ^ ^ ' 2x + a * or (^2x -\-d){dp- dq) + A,pdx = o (5) To ascertain whether this is an exact equation, subtract from the first member the differential of {2x -\- d) (p — q), which is {2x -\- a) {dp — dq) -f 2pdx — 2qdx. I I § XXIII.] EXAMPLES OF MONGERS METHOD. 339 The remainder is 2pdx + 2qdx, which, since dx = dy, is equivalent to 2d2. Hence, equation (5) is exact, and gives the integral {2x + a){p-q) + 2Z=^b (6) From equations (4) and (6) we have the intermediate integral {x-\-y){p-q) + 2Z^<^{y-x) (7) Lagrange's equations now are dx _ dy _ dz x-\-y X -{- y (ji{y — x) — 2Z whence we have the integral x+y = a, (8) which converts the relation between dy and dz into dz 2S _ {2y — a) dy a a The integral of this last equation is 2^ " = --J^ '' {2y-a)dy + p.. .... (9) Finally using equation (8) and putting /8 = -«/'(a), we have a __2y_ - _2y {x-\-y)ze -^y = -\e <^ {2y-a)dy-\-xl;{x+y), . (10) : where ;r + j is to be put for a after the indicated integration. 312. In this example it was not possible to obtain the second integral required in Lagrange's process in a form containing a simple arbitrary function of the form <^(z^), as was done in finding equation (9), Art. 309. Thus the final integral in the present 340 EQUATIONS OF THE SECOND ORDER. [Art. 3 1 2. case is not of the form considered in Art. 300. In the case of a primitive of the present kind, there is but one intermediate integral. Accordingly, it will be found that, had we employed the other factor of equation (2), the resulting system of Monge's equations would not have been integrable. Examples XXIII. Solve the following partial differential equations : — I. r^f{x,y), z ^\[f{x,y)dx^^x^{y) + x\,{y) z = ^x^ logy + axy + <^(^) + xl/{y) z=y{ey — e^) + {x) + eyxlf{x) z = ^x^'y — xy + cf>(y) + e-^i{/(y) z = ^x^y + (y) log;(f + if/{y) z^ = x^y^ + x(y) -\- ilf{y) z = log le^ycf>{y) - e-^y'\ + y\,{y) z = {x)il/{y)x^°sy z—{x +y) logy + <^{x) + i/'(^ + y) x=ct>{z) +x(,{y) 11. x^r -h 2xys -\r y^^ = o, ^ =^ "^^^ ( ^ ) + '^ (^j 12. r — a^t—o, z — f^{y ■\- ax) -\- \^{y — ax) 13. ^V - y-'t =qy-px, z = I -J -\- if;{xy) 14. ^(i +^)r-(/ + ^ + 2/^)j-+/(i +/)/=o, xz= ct>{z) + i(/{x -^y + z) 15. (3 + ^^)^r — 2(^ -f- ^g) {a + c/>)s + {a-h cpyt = o, y -\- x{ax + 3y -\- cz) = ^{ax + dy + cz) 2. 3- t-q = e^ J^ey, 4- p + r^xy, 5- xr + p=. xy, 6. zr+p-=7,xy^, 7- r+P'=y^ 8. z^ 9- c. ps — qr = 0, § XXIV.] LINEAR EQUATIONS. 34 j XXIV. Linear Equatioiis. 313. A partial differential equation which is linear with re- spect to the independent variable z and its derivatives may be written in the symbolic form F{D,D')z=V, (i) where dx dy and F is a function of x and y. We have occasion to consider solutions only in the form z^f{x,y), and shall therefore speak of a value of z which satisfies equa- tion (i) as an integral. Since the result of operating with F{p, U) upon the sum of several functions of x and y is obvi- ously the sum of the results of operating upon the functions separately, the sum of a particular integral of equation (i) and the most general integral of F{D,D^)z = o (2) will constitute the general integral of equation (i). Hence, as in the case of ordinary differential equations, the general in- tegral of equation (2) is called the complementary function for equation (i). So also, as in the case of ordinary differential equations, when the second member is zero, the product of an integral and an arbitrary constant is also an integral ; but this does not, as in the former case, lead to a term of the general integral, since 342 HOMOGENEOUS LINEAR EQUATIONS. . [Art. 3 1 3. such a term should contain an arbitrary function. It is, in fact, only in special cases that the general integral consists of sepa- rate terms involving arbitrary functions. Homogeneous Equations with Constant Coefficients, 314. The simplest case is that in which the equation is of the form Ao- V A^- + . . . + ^„-— = o, . . . (i) doc^ dx^'-'^dy dy» the derivatives contained being all of the same order, and their coefficients being constants. Let us assume 2 = ^ (jv + mx) . Now — - xpiy -f- mx) = m\l;'{y + mx) and — - ^{y + mx)=\l/'(y + mx), ax ay whatever be the form of the function xjj, therefore the result of substitution, after rejecting the common factor <^^*^^{y'{-mx)y will be Aofn*' + A^m^-'' + . . . + An — o (2) Hence, if w be a root of this equation, 3 = <^(j -h mx) satisfies equation (i), <^ being an arbitrary function. If m^, m^, . . ., ^n^ are distinct roots of equation (2), we have the general integral z=^x{y + m^x) -+- i, <^2, . . ., <^« are arbitrary functions. Given, for example, the equation d^z d'^z , , d^z 3a — — -f 2^2 — - = o. dx"^ dxdy dy^ The equation for m is m^ — ^am -f- 2^= = o. § XXIV.] WITH CONSTANT COEFFICIENTS. 343 whence m=za or m = 2a. Hence the general integral is z= {y -\-ax) +\l/{y -\- 2ax). 315. Equation (i) of the preceding article, when written symbolically, is {AoD^ + A^D^-^D' + . . . + A„D^^)z = o, or, resolving into symbolic factors, (Z) - m^D^){D - m^D') . . . {D - mnD')z = o. . . (4) Since the factors are commutative, this equation is evidently satisfied by the integrals of the several equations, {D-m,D')z = o, {JD - m^D^)z = o, ... {D - mnD')z = o. Accordingly the several terms of the general integral (3) are the integrals of these separate equations. Again, the equation may be written ^'"/(|,)-=°. • (5) where/ is an algebraic function of the nth. degree, and equation (2) is equivalent to f{m) = o._ We may now regard the symbol — , when operating upon a func- tion of the form <^(j/ + mx) as equivalent to the multiplier m, thus It follows that j^My + ^^) = ^<^(i' + ^^)' -^\D' ' *^^^ ^ ^^^ =/(^) 344 LINEAR EQUATIONS. [Art. 3 1 5. SO that equation (5) is satisfied by ^{y -f- mx) when f(m) — o, whatever be the form of the function <^. 316. The solution of the component equations, of which the form is {D- jnD')z = o (i) may be symbolically derived from that of the corresponding case of ordinary differential equations. For, if we regard D' in equation (i) as constant, its integral is where C is a constant of integration. Replacing C by <^(/), as usual in integrating with respect to one variable only, we have for the symbolic solution — ^mxD' Hy), (2) where <^(j) is written after the symbol because D' operates upon it, though it does not operate upon x. The symbol e""^^' is to be interpreted exactly as if D' were an algebraic quantity. Thus = 4,{y) + mx^\y) +^^<'(y) + . . ., 2 ! or e**"^^'€fi{y) — (^{y -\- mx), by Taylor's theorem, of which this is in fact the symbolic state- ment (Diff. Calc. Art. 176). It should be noticed that the process of verifying the identity {D - mD')e^^D'{y) = o, § XXIV.] CASE OF EQUAL ROOTS. 345 with the expanded form of the symbol e*"^^', is precisely the same as that of verifying {D — m)e"^^ = o, with the expanded form of the exponential e"^"^. Case of Equal Roots. 317. The general solution, equation (3), Art. 314, contains n arbitrary functions ; but when two of the roots oif{m) = o are equal, say m^^ = niz, the corresponding terms, are equivalent to a single arbitrary function. There is, how- ever, in this case also, a general integral containing n arbitrary functions. To obtain it we need an integral of {Z>-m,D')'z=o, ....... (i) in addition to that which also satisfies (B — m^U)z = o. This required integral will be the solution of {D - m^D^)z ^ <^{y -^ m^x) ; (2) for, if we operate with D — m^D^ on both members of this equa- tion, we obtain equation (i), so that its integral is also an in- tegral of equation (i). Writing equation (2) in the form p — m^q= ^{y-\-m^x), Lagrange's equations are ^^=_!Si= ^ , 346 LINEAR EQUATIONS. [Art. 3 1 7. of which the first gives the integral y -j- tn^x = a, and then the relation between dx and dz gives z = x{a) + 3. Thus the integral of equation (2) is z — x{y + m^x) -f il/{y + m^x) ; and, regarding <^ and «A as both arbitrary, this is the general inte- gral of equation (i). 318. The solution may also be derived symbolically ; for, since the solution of (D — nifz — o is we have, for the solution of (Z> — mDyz = 0, z^e^^i>'\_xf^{y)^-y\,{y)\ that is, z^= x^^i^y -Y mx) -^-y^i^y -\- mx) (1) The solution might also have been found in the form z=y^x{y -^mx)-\-\\i^{y-\rnix), (2) but this is equivalent to the preceding result ; for we may write it in the form ^ =-\y ^-nix — mx)i(y + mx) -h «/'i(>' + mx) ; and, since (jy + mx) i(jy + mx) -\-^i(y + mx) and — m4>i{y 4- mx) § XXIV.] CASE OF IMAGINARY ROOTS. 347 are two independent arbitrary functions of y-\-mx, they may be represented by \p and <^, the equation thus becoming identical with equation (i). In like manner, if the equation /( — J = o has r equal roots, the terms corresponding to {D — mUy are Case of Imaginary Roots, 319. When the equation has a pair of imaginary roots, /t ± ivy the corresponding terms in the general integral are z = (y + fjLx -\- ivx) + \p{y + fix — ivx) ; or, putting u=zy-\-fix, v — vx, <^(« + iv) + «/'(« — iv). To reduce this expression to a real form, assume so that <^ = i( and ^ = ^ (i - i^i) * Making the substitutions, the expression becomes 2 = i-[<^i(« + t'y) + ', which is readily verified. The Particular Integral, 320. The methods explained in the preceding articles enable us to find the complementary function for an equation of the form F(^D,D^)z^V, when F{p^ D^) is a homogeneous function of D and D\ and Fa function of x and y. The particular integral, which is denoted by I V can also in this case be readily found. Resolving the homogeneous symbol F{Df D') into factors, we may write F(D, £>') = (Z) - m,D') {D - m^) . . . {D - ninD'), § XXIV.] THE PARTICULAR INTEGRAL. 349 and the inverse symbol may be separated, as in Art. 105, into partial fractions of the form where the numerators are numerical quantities, and ;' is unity except when multiple roots occur. It is therefore only neces- sary to interpret the symbol 321. For this purpose we employ the formula proved in Art. 116. Putting mD' in place of a,* this formula gives =■ e^^^' — ^{x, y — mx) (i) Hence the result is found by subtracting mx from j^ in the oper- and, integrating with respect to x, and adding mx to y after the integration. Since * In explanation of this application of the symbolic method, let it be noticed that, just as the formula of Art. 116 is founded upon the equation De^^ V^ e^^{D + a) V, so the present application of it depends upon De*"^^'^{x, y) = e^^'^XO + wZ)') ^{x, y), or Di(x, y -h mx) = result of putting y + mx for _y in (Z> + mD') ^(x, y)y which expresses an obvious truth. 350 LINEAR EQUATIONS. [Art. 32 1. j'^(^, y - mx)dx = f^H, y - mi)d^, this may be expressed by the equation £> -mD' ^^^' ^^ ^ r*^^^' y + mx- mi)di. ... (2) In like manner, for the terms corresponding to multiple roots olf{m) = o, we have (^ ;^;^,^(^, J^) = j"J. . >^^{k,y^mx - mi)di^, . (3) 322. There are certain methods by which, in the case of special forms of the operand, the result may be obtained more expeditiously than by the general method just given. Some of these, which apply as well when the equation is not homogene- ous, will be found in Arts. 328-334. The following applies only when the equation is homogeneous. Suppose the seco7id member to be of the form, ^{ax + by). The equation may be written in the form F{D, D')z = ^y f^') 2 = ^(ax + by). It is readily seen that f(^§^^(^^ + by) =/g) H^^ + by). We have, therefore, for the particular integral or, denoting ax + by by t, since a"f[-] =-^{^> ^)> § XXIV.] SECOND MEMBER OF THE FORM ^ {ax -\- by). 35 I ^=^fl- ••!*«'''"•• ^^> Given, for example, the equation 2 — = sm ix + 2v) , dx^ dxdy dy^ ^ -^^' the particular integral is — sin {x + 2^) X)2 _|_ 2)/)' _ 2Z>'^ = ^- sin tdt^ = i sin/ = i sin (;t: + 2^). 1 + 2 — 8JJ 5 5 Adding the complementary function, z = (^(^y 4- ^) + ^{y — 2^) + ^ sin {x + 2y), 323. When i^(^*^+^ = he^^'+^y and He^^'-^^y = ke^^+*y, cF{h, k)e^^-^f'y=zo. Thus we have a solution of the assumed form, if h and k satisfy the relation J'{h,k)^o, (3) c being arbitrary. Let equation (3) be solved for h in terms of k. Now if F{hy k\ is homogeneous, we shall have roots of the form h = m^kf h = m^ky . . ., h = m„k ; § XXIV.] THE NON-HOMOGENEOUS EQUATION. 353 and, since the sum of any number of terms of the form (2) which satisfy the condition (3) is also a solution, the equation will be satisfied by any expression of the form where m has any one of the values m^, m^y . . ., m„. But, since for a given value of m this expression is a series of powers of ^y+mx with arbitrary coefficients and exponents, it is equivalent to an arbitrary function of ^>'+'«-^, that is to say, it denotes an arbitrary function oi y -\- mx. This agrees with the result other- wise found in Art. 314. 325. Again, if F{D, U) can be resolved into factors, and one of these is of the form D — mH — by so that F{ky ^) = o is satisfied by h = mk + b, equation (i) will be satisfied by an expression of the form z = %ce^^y + '«-^) + ^^, where m and b are fixed and c and k are arbitrary. But this ex- pression is equivalent to the product of ^^-^^ into an arbitrary function of 7 + mx. Thus, corresponding to every factor of the form D — mD — ^ we have a solution of the form z = e^^<^{y + mx). Given, for example, the equation d^z _ d^ I ^ ^ _- dx"^ dy"^ dx dy * or the general integral is 354 LINEAR EQUATIONS. [Art. 325. We might also have found the solution in the fori z=^ ^,{y - X) ^ eyy\,^{y -^ X) ', but, writing the last term in the form ey^^~*\\i^{y -\- x)^ this agrees with the previous result if «/'(/) is put for e^\p^{t). 326. In the general case, however, we can only express the solution of F{D,D')z = o (i) in the form z = ^ce'^^ + ^y, (2) where F{h, k) = o, (3) so that c and one of the two quantities h and k admit of an infinite variety of arbitrary values. Given, for example, the equation d^z dz _ dx^ dy Here F{p, U) = Z>' — D\ whence k^ — k — o, thus the general integral is z == '^ce''^^ + ^'^y. Putting h-= I, h — 2, k = \, etc., we have the particular integrals e^+y^ e'^-^^y ^i^+b', etc. Special Forms of the Integral, 327. There are certain forms of the integral of F(Z>, D^)z = o which can only be regarded as included in the general expres- sion (2), by supposing two or more of the exponentials to become identical. Let the value of k derived from F{liy /^) = o be k^Ah), (4) then hi — ^2 § XXIV.] SPECIAL FORM^ OF THE INTEGRAL. 355 is an integral of FiDy D')z = o. When h^^h^-Bs. h^ this takes the indeterminate form, and its value is dh which is accordingly an integral. In like manner we can show that — ^>^-^+/(^):»', and in general, — 1-^A-^+/(A):»' satisfies equation dk' dk- ^ (i) ; thus we have the series of integrals .(5) For example, in the case of the equation (D^ — D');s = o, the integral ,?'^-*+^'y gives rise to the integrals e^^ + ^'y (x-^ 2hy), /■^ + ^''^[(^ 4- 2kyy + 6y(x + 2kyy], e^^ + ^'>'[{x + 2ky)^ + i2y{x + 2/iy)^ + J2y^2f In particular, putting /i = o, we have the algebraic integral z — CiX-\- c^^x"^ 4- 2y) + c^{x'i -\- 6xy) Special Methods for the Particular Integral. 328. The particular integral of the equation 356 LINEAR EQUATIONS. [Art. 328. is readily found in the case of certain special forms of the func- tion V. In the first place, suppose V to be of the form e^^-^^y. Since De^^^h^ae^'^'^^y and D'c^'^^^y ^bc^'^'^^y, and FiDy D') consists of terms of the form D'^U^ we have F(^D, Z>') ^-^ + ^^ = F{a, b) ^«^ + h^ or ^ F(a, b)e''^ + ^y = e^^ + h where Fifty b) is a constant. Hence, except when F{ay b) = o, we have . ^ax + iy ^z ^-^ + ^y. F{Dy D^) F{a, b) Thus, when the operand is of the form e'^^'^^y^ we may put a for D and b for D\ except when the result introduces an infinite co- efficient. Given, for example, the equation the particular integral is Z = ^ ^2j: + J/ _ 1 ^2jr + >, 329. In the exceptional case when F{a, b) = o, we may pro- ceed as in Art. no. Thus, first changing a in the operand to a + h, we have Z= ^x+Ax+iy^ 1 gax+3yfj^/ix^^^^ _\ F{D,F>') F{a-\-h,b) \ 2 ) The first term of this development is included in the comple- mentary function. Omitting it, we may therefore write for the particular integral § XXIV.] SECOND MEMBER OF THE FORM e^^-^h, 357 {x-\-\hx-^ -^ . . .)e^^ + ^y, F{a + A,d) in which the coefficient takes the indeterminate form when /i = o, because F(a,d)=o, and its value is — — — -, where FJia, b) ^a\(i, 0) denotes the derivative of Fia, b) with respect to a. Hence, except when Fa {a, b) = o, we have e'^^ + h (i) FJ{a, b) In Uke manner, if Faia, b) = o, the second term of the de- velopment is in the complementary function, and we proceed to the third term. It is evident that we might also have obtained the particular integral when F{ay <5) = o in the form eax + by.^ ........ (2) Fi\a, b) but the two results agree, for their difference, J_1 _FJ{a, b) F,\a, b)] gax + by^ is readily seen to be included in the first of the special forms (5) of Art. 327, since a and b are admissible values of the h and k of that article. 330. In the next place, let V be of the form sin {ax + by) or cos {ax-\-by). We may proceed as in Arts. 11 1 and 112, and it is to be noticed that we have, for these forms of the operand, not only V^ — a"" and n"" = — b"", but also DD^ = — ab. Given, for example, the equation d'^z . d^^z , dz • / , \ 35^ LINEAR EQUATIONS. [Art. 330. the particular integral is z — sin {x + 2y) — — ^ — sin {x + 2y) D^-\-DD'-^D' -\ ' ^' Z>'-4 /)' + 4 sin (jc + 2y) = — tV [cos (^ -h 27) + 2 sin {x + 2jv)] . Z>'»- 16 Adding the complementary function, we have z = e^^i^y — x) -\- e-^\p{y) — j\ cos {x + 2j;) — -J- sin (x -f 2>'). The anomalous case in which an infinite coefficient arises may be treated like the corresponding case in ordinary differen- tial equations. 331. Again /et V be of the form xy% where r and s are positive integers. In this case, we develop the inverse symbol in ascending powers of D and D'. Thus, if the second member in the example of the preceding article had contained the term x^yy the corresponding part of the particular integral would have been found as follows : z == x^y i-{D^Jf. DV + D') ^ = - [i + (^^ + DD' + D') + {D^ + DD* + ny + . . ."Ixy = - [i -f- Z)^ -f DD' -f Z)' + 2D^D''\ x^y St — x'^y — 2y — 2x — X' — 4. It will be noticed that, on account of the form of the operand, it is unnecessary to retain in the development any terms containing higher powers than Z>^ and D'. Again, had the operand been xy, we might have rejected D^ in the denominator thus : ^y= 7:^. — —^^y -[i4-Z>'(i +Z>)]^j'= -xy-x— I. I § XXIV.] SECOND MEMBER OF THE FORM x^f. 359 332. When the symbol F{p, Z?') contains no absolute term, we expand the inverse symbol in ascending powers of either D or D\ first dividing the denominator by the term containing the lowest power of the selected symbol. For example, given the equation dx^ dxdy for the particular integral we have to evaluate In this case, it is best to develop in ascending powers of D\ because, with the given operand, a higher power of D than of D^ would have to be retained. Thus ^:p^)--=iC-^3f)-. D^ D^ 12 20 Adding the complementary function, 2 = <^ W + ^{y + 3^) + -hx^y + i^x\ If we develop the symbol in ascending powers of Z?, the par- ticular integral found will be _ x^y"^ x^y"^ xy^ ~ 18 54 324* The difference between the two particular integrals will be found to be TiE^li3x+yy-y'2, which is included in the complementary function. 36o LINEAR EQUATIONS. [Art. 333. 333. Finally, whefi the operand is of the form e^'^^yVy we may employ the formula of reduction F{D, D')e^^ + ^yV= e^'^hF{D -\- a, D' -\- b) V, which is simply a double application of the formula of Art. 116. For example, _ gax +a?y _J I ^ 2aD ^ D^ 2a 2a 2a D\ 2a J \^a ^a^J If we develop in powers of D\ we shall find ^ xe^^+'^^y = — e^x + '*''y(xy -\- ay'). The difference between the two results is accounted for by the special forms given in Art. 327 for the complementary function in this example. 334. As another application of the formula, let us solve the equation dx* dxdy dy The particular integral is z = the coefficient of; in ___^^_^^^-+^>i^. Now gix + lyx^ — H- DD^ - 6X>'=^ {D + iy + i{D + /) - 6; = I -^gtx + iy. D^ + ZiD + A § XXIV.] SECOND MEMBER OF THE FORM e^^' + hy. 36 1 and by development we find therefore I D^-VDD'-dD'^ gix + tyx^ = [cos {x +)>)•{' i sin {x + j;)] T— - 3^ - ill L4 8 32J Taking the coefficient of /, and adding the complementary function, 2=( — - — )sin(^^->')-^cos(:v+>')4-<^(>'+2^) + ^/'(J-3^)• \4 32/ « Linear Equations with Variable Coefficients. 335. In some cases a linear equation with variable coeffi- cients can be reduced, by a change of the independent variables, to a form in which the coefficients are constant. As an illus- tration, let us take the equation 2_ d^ L ^ — JL ^ L (^ ( \ x^ dx^ ■x3 dx y^ dy^ y"^ dy The first member may be written in the form I fi ^^2 i^ dz ~\ _\ d I X \jc dx^ x^ dxj X dx x dz_ dx Hence, if we put xdx = d^^ whence I = \x'^y and in like manner V = \y^i the equation becomes dt~ dff ^^^ The integral of this equation is ^ = <3!>(|+ r;) + 1/^(^— ■>;) ; hence that of equation (i) is z = {x^ -iry^) + xIj{x^ -r)- 362 LINEAR EQUATIONS. [Art. 336. 336. In particular, it is to be noticed that an equation all of whose terms are of the form Ax^'y'- — — is reducible to the form with constant coefficients, like the cor- responding case in ordinary equations, Art. 123, by the trans- formations ^ = log ;»r, 17 =s log/, which give dx di dy d-q But, if we put t? = X—- and i9' = j^ — -, we may still regard x and y dx dy as the independent variables ; the transformation is then effected by the formula dx^'dy^ and the equation reduced to the form F{J^, ^^)z = V, The solution of this equation may therefore be derived from that of the equation F{Dy D') .s" = F, by replacing x and y by log;ir and logy \ or it may, as in the following articles, be ob- tained directly by processes similar to those employed in deriv- ing the solution of F{D, D')s = V. 337. Since ^x^y = rx'^f, ^^x'^y* = sx^y^, it is obvious that F(^^,^')x^y^ = F{r,s)x''y. ...... (i) § XXIV.] THE EQUATION F(J^, ^') z = O. 363 Hence, if in F{^9, ^')z = 0, (2) we assume z = cxy*f the result is {:F{r, s)x^y^ = o, and we have a solution of the proposed form if F(ry s) = o. Hence the general solution of equation (2) is 2 = 2^^0^, (3) where ^(^ ^) = o, (4) that is, 2^ is a series in which the coefficients are arbitrary, and the exponents of x and _>/ are connected by the single relation (4). Now let equation (4) be solved for r in terms of s; if the function F(^, ^') be homogeneous in ^^ and t?', the equation will, have roots of the form r = m^s, r = m^s, etc., and to each root will correspond a solution of the form y — %c{yx'*'y. But this represents an arbitrary function of yx^. Thus to each factor of f\^, ^') of the form there corresponds an independent term of the form z = 4>{yx^) in the solution of equation (2). 364 LINEAR EQUATIONS. [Art. 337 Again, corresponding to a factor of the form ^ - m^V - b, we have the root r = wj + by for F{ry s) = o\ and hence the solu- tion s = ^ijx*"yx^y or 338. For the particular integral of the equation we may suppose V to be expanded in products of powers of x and J/. By equation (i) of the preceding article, we have which gives the particular integral, except when F{a, d).= o. When this is the case, we have, first putting a + k in place of a^ 1 — ^^a+Ay = 1 ^y(i ^ klosx + . . .), or, rejecting the first term of the expansion, which is included in the complementary function, and then putting k = o, jc«y =z= x^y^ log X. /^(^'^') ^ Fa\a,b) ^ ^ 339. As an illustration, let us take the equation x^ V* — = xy. dx^ -^ dy^ '^' which, when reduced to the t^-form, is ,^(,?_ 1)2 _,:/'(/>'_ i)z = xy, § XXIV.] THE EQUATION F(^^,f^^)z=iV. 365 or The coipplementary function is for the particular integral. I II xy = xy or, rejecting the term —xy included in ^{xy), and putting ^ = o, 2 = <^{xy) -\- xy\i[±\ -\- xy logjxr. ^O 340. The symbol t^ + '*^' may be particularly mentioned on account of its relation to the homogeneous function of x and y. Putting we have irx'^y^ = {r-\-s)xy^ ; hence, if u„ denotes a homogeneous function of x and y of the nth degree, we have 7rU„ = nu„, where u„ is not necessarily an algebraic function, but may be any function of the form x"fii-i This is, in fact, the first of Euler's theorem concerning homogeneous functions. See Diff. Calc, Art. 412. As an example of an equation expressible by means of the single symbol tt, let us take 366 LINEAR EQUATIONS. [Art. 34O. x*'-— + nx'*--'y- T + — ^-x»-'y'- — 4-... = K (i) ^x** dx^-^dy 2 dx^-^dy^ The first member can be shown to be equivalent to 7r(7r — l) . . . (tt — « + i)z. Denoting this by F{Tr)z, we have F{Tr)u^ — m{m—i).,.{m^n+i)u^, . . .(2) which, when F(Tr) is expressed as in equation (i), is the general case of Euler's theorem. Thus the complementary function for equation (i) is Uo-\-Ur + u^ + , . .-{- U„-r. Let F contain the given homogeneous function Hf„, equation (2) gives for the corresponding term in the particular integral m{m — i) . . . {m — n -\- i) except when m is an integer less than ;/. In this case F{Tr) will contain the factor Tr — m, and putting F{ir) = {Tr — m)(Tr) we readily obtain as in Art. 338 Examples XXIV. Solve the following partial differential equations : — d^ z d^z d^z d^z d^z , d^z I 2. 2 1 = —•) dx^dy dxdy^ dy^ x^ ' z == <^(^) ^if,(x-}-y) + xxix -\-y) — j'log^. § XXIV.] EXAMPLES. 367 dx^ dxdy dy^ y — 2x' ' z — {y — 2x) + xp{y — 3^) -f. ^ log {y — ix). d'z d^z ■ d^z dz dz _ dx^ dxdy dy^ dx dy z = (x) +£-^^ily{y) + {m -\- a){n + b) d^z d^z , dz / , \ , V 7- TT - -1—7 + ~r - z = cos{x -\- 2y) + e^, dx^ dxdy dy zz=z e^ (jii^y) -{- e- ■^ ^{x -\- y) -\- ^ ^\n{x + 2y) — xey. o d'^z d^z , dz dz • / , \ 8. = 2 sin(^ -\- y)i dx^ dxdy dx dy v ^-^/' z — e-''<\>{y^-\-\\i{x-{-y^-\-x\%v!\{x-\-y^— cos {x-\-y)'\. dz dz ^^ 9. a — = e**^"^ cos ny, dx dy z=. rf) ( y + ax) H (m cos ny — na sin m^) . m^ + n'^a^^ d'^z d^z , dz . dz ^ « „ dx^ dy"" dx dy z = e^4>{y — x)-\- e-^'^^if/ {y + x) -ie^-y + ixy-\-^x- + 4^xy + ix + y + ^. 368 LINEAR EQUATIONS. [Art. 340. II. mn (;//' + w=) + mn——-\- mn^ tn^n — dx"^ dxdy dy^ dx dy = cos {kx + /f) + cos {mx + ny), zss(ny + mx) + e-»^ij/{my + «^) ffl;gsin(^j[: + i^) — (ffl/^ — ;g/)cos(^.y + /v) (;2/& — m/) [m^n^ + {mk — niy] mnx cos{mx + ny) + {m'^ — ;g').y sin(;^;c + ny) ,^«2 , ^=*2 , ^d^z dz dz , 13. x'-r- -\'2xy——---^y^- nx- ny—- + nz = o, dx'' dxdy dy^ dx dy d^ z dz f 14. (^+^)^-^^ = °' 2 = \(x+yy'(^)dx + ^{y)' d^z , d^z v-xf \ 16. Derive the particular integral of d^z , d^z ^_xf , \ in the form z = ^_>' -^ ^ Re^wfid l^cwks ateswbjtctfto r^l4ied|itc recaiL 6 9 Ju n ^ 6 0V" REC'D LD MAY 27 I960 lygCfTft'tK APR 8 - 1956 5 NIAR2 9 195S14 OCT 17 1972 3 LD 21A-50to-4,'60 (A9562sl0)476B General Libruy Unlvenity of California Berkeley '■■■-■ -•' m!\ &:■:■'■■» . • ^ :■*. ■ "' d a UNIVERSITY OF CALIFORNIA LIBRARY ii^>J*!4